Acta Mechanica 154, 129 140 (2002) ACTA MECHANICA �9 Springer-Verlag 2002

Finite e lement analysis of thermopiezoelectr ic smart structures

A. G6rnandt and U. Gabbert, Magdeburg, Germany

(Received December 19, 2000)

Summary. In many applications of piezoelectrically controlled smart structures, the functionality of the system has to be ensured even in an extremely hot and cold environment. Hence, the thermal effect is very important and must be taken into account when designing such structures. To this end, a fully coupled formulation of the piezothermoelastic material behavior is required. Based on a weak formulation of the equilibrium conditions and the coupled constitutive equations of this three-field problem, a general finite element concept is presented to solve such problems numerically. Both the static and the dynamic response of combined thermal, electric and mechanical excitations are considered. An iterative solution strategy is proposed to solve the coupled field problem in an effective way.

1 Introduction

The increasing activities in the development and industrial application of smart structures

require effective simulation and design tools. For vibration, noise and shape control piezoelec- tric wafers and fibres are mainly used as actuators and sensors in such highly integrated smart structures. Although the piezoelectric effect has been known for a long time (J. and P. Curie, 1880), its use in the control of systems is fairly new (see, e.g. [1]). Several classical textbooks

describe the theoretical foundations of the piezoelectric effect as a coupled field problem

([21 - [6]), and analytical solutions are given to solve engineering problems ([7], [8]). However, the analysis and the design process of engineering smart structures with integrated piezoelec-

tric actuators or sensors require powerful calculation tools. Here, the finite element method (FEM) provides an effective technique. Due to its wide-spread use, a lot of theoretical and

practical results are available from a wide range of applications. This method can also be employed for solving coupled piezoelectric field problems. Over the past few years consider~ able progress has been made in developing finite elements for coupled electromechanical fields. But many of such software developments have been restricted to simple applications such as beam and plate structures ([8], [9]). Only recently, our group extended a commercially available finite element software by including a pool of piezoelectric finite elements to provide

solutions for 1D, 2D, 3D and layered composite shell problems as well as numerical solution techniques for static and dynamic problems, optimization algorithms (e.g., for an optimal pla- cement of active materials in a structure) and control algorithms (including a qualified data interface to special control design tools, such as Matlab/Simulink) [10], [11].

In several applications of smart structures thermal effects play an important role and have to be taken into account, e.g., in aerospace structures. Although the thermal aspect is included in some finite element developments ([12]-[16], etc.), most of them only consider tempera-

130 A. G6rnandt and U. Gabbert

ture-induced deformations and the pyroelectric effect. This paper aims at reviewing the fully coupled thermopiezoelectric field equations to provide a basis for finite element discretization and solving these equations numerically. It is shown that the balance laws of mechanics and electrodynamics together with the entropy production inequality and the Gibbs relation are a suitable basis for linear thermopiezoelectric finite elements. The finite element equations are developed in a general manner, independent of a special finite element type. To solve coupled field equations in dynamics numerically, an iterative solution strategy is used which seems to be more effective than the solution of the fully coupled non-symmetric equations. An example demonstrates that it is required to take into account the thermal effect on piezoelectrically controlled smart structures.

2 Linear theory of thermopiezoelectricity

2.1 Balance equations, first and second principle o f thermodynamics

The electromagnetic field can be described by the four Maxwell equations [17]:

e i j kHk , j = Ji + f ) i , e i j kEk , j = --J3i , Bi , i = 0 , Di,i = Pc , (1)

with the five vector fields of Ei - vector of the electric field, Di - vector of the electric displace- ment, Hi - vector of the magnetic field, Bi - vector of the magnetic induction, J / - vector of the conduction current, and e/jk, & as the permutation symbol and the free electric charge, respectively. By introduction of the scalar electric potential qo and the vectorial magnetic potential Ai, the Maxwell equations can be replaced by the following system of differential equations [18]:

Be -- e/~Ak,j, Ek = --~,~ -- Ak, eijkHk,3 = J~ + D/, D/~/= &. (2)

For piezoelectrics, i.e., polarizable but not magnetizable dielectrics, the assumptions & = J / = 0 are valid [7]. The basic simplification ]A/] << Iqo/] is fundamentally valid for elec- tromagnetic waves, which are not coupled with the elastic waves, and for wavelengths near those of elastic waves [7]. With this assumptions the Eqs. (2) simplifiy to

c/jkHk,j = D/, E / = -~ , / , De,/= 0. (3)

The complete energy balance of a piezoelectric body with the volume V and the surface O can be expressed by

d ( o U + K ) d V = L + d~ (4)

y

U and 8 are the internal energy-density and the mass density, respectively. K is the kinetic energy-density

K = ~ v~vj, (5)

where vj is the vector of the displacement velocity. L is the power of the external mechanical and electric forces,

L = f & v j dV + f,~/(T~jv 5 - ~be) dO, (6) v o

Thermopiezoelectric smart structures ! 31

where Tij = Tji, bi and n~ are the symmetric Cauchy stress tensor, the vector of body forces and the normal vector to the surface O, respectively. The last term of Eq. (4) is the non- electromechanical power [19]

d~ = Or d V - niq~ dO, (7)

V 0

where q~ and r are the vector of heat flux and the quantity of heat generated by internal heat sources per unit mass and unit of time, respectively. With the divergence theorem and Eqs. (5)-(7), Eq. (4) can be reformulated as

f o O d v : I(T~j,~ + ob - o+,)~ d v + f (~y s ,~ - ~,J)~ - q,:,~ + o ~ ) d r . (S) V V V

Under consideration of the local form of the equations of motion

Tij,i + @bj - - s = O , (9)

the balance of the inner energy results in

f (oO - ~v j ,~ + < j ) ~ + q~,~ - o~) dV = O, (lO) V

or in local form

O 0 = T i j v j , i - ~ , i b i - qi,i -k Or . (11)

The second principle of thermodynamics can be written in the form of the local Ctausius- Duhem inequality [19]:

~o//+ (q~) - ( - ~ ) >_ O, (12) ,i

with the specific entropy T/and the absolute temperature (3. The application of the quotient rule results in

Of l q qi,i(3 - qi(~,i or > 0. (13) (32 ( 9 -

2.2 Consti tut ive equations

In addition to the balance equations, the constitutive equations are required for the descrip- tion of the individual thermopiezoelectric behavior of different materials. The displacement gradient uj,i, the vector of the electric field E~, the absolute temperature (3 as well as the spa- tial gradients E<j and the O~ are chosen as independent variables. Then, the stress tensor Tij,

the vector of the electric displacement Di, and the specific entropy r/are dependent variables. This is a suitable, but not the only possible way of selecting dependent and independent vari- ables. The electric Gibbs potential 0G = @G(uLi, Ei, E{,j, 0, I~j) can be written as [3]

oG = oU - E~D{ - (3Q~7. (14)

Differentiation of Eq. (14) with respect to time gives

OG = O0 - Z{bi - D{E{ - ~)~i7 - ~n6. (15)

132 A. G6rnandt and U. Gabbert

The total time derivative of ~G(uj,i, ~i, Ei,j, ~ , (~,i) results in

OG . OG OG OG OG (16)

From Eqs. (15) and (16), taking into account Eq. (11), follows

�9 ( 4

< ) (os + 4 ~ + f ] O + 0 ( ~ , i + @ O i l + q i , i - o r = O . (17)

Equation (17) must hold for all admissible values of ga,i,/~i,/~,j, 0, 0 i , and, hence, the expressions in parentheses must be vanishing identically, as these expressions are independent of the time derivatives. This results in

OG OG OG OG

so that the electric Gibbs potential is independent from the spatial gradients Ei,j and (3~. Further, the remaining energy balance results from Eqs. (17) and (18):

O~oil + q.i,{ - or = 0. (19)

Introducing ~9 = (3 - O0 as the temperature change with respect to a stress free tempera- ture (30 and expanding the electric Gibbs potential around the reference state 4G(uj,i, El, 0 )

= 4G(O, 0, O0) gives

OG u OO E.~+4 ~G(%~, & , O) = 40(0, O, (30) + ~ 0%~ o J'~ + 4 ~

1 ( 02G 02G 02G 2"~

.4 + 4 UoE oe E, + ~ o ) 0 2 G { 02G 02G

+ 4 Ouj,~OEklo uj,~Ek + 0 ~ o uj,~O + 0 O ~ O 0 o &~) + ~ (2O)

If the existence of a natural state corrresponding to the reference state is assumed, where

QG(0, 0, @0) = 0 and Tij(0, 0, O0) = Di(0, 0, @0) = r/(0, 0, @0) = 0, the linear terms of the Taylor series vanish by virtue of Eq. (l 8) [20]. The quadratic terms correspond to the constitu- tive equations of a geometrically and physically linear material theory. Their coefficients are

denoted as follows:

02G _ OT~ A _ OD~ _ 02G OTij _ Of] _ 4 0uj,~ OEk OEk Ouj,i ek~j, 4 Ouj,i 0 0 -- OO Q Ouj,i ( i j , (21)

02G ODi _ Or 1 02G OTij _ C i y , (22) OE~ 06) - 0 0 ~ - ~ i = -P~ ' ~ Oui,~ Ou~,l - Ouk,z

02G OD~ = c92G 0r] (23) o OE~ OE 5 = - OEj - ~ J ' o 0(35 - 4 ~-~ = - a .

As cv, the heat capacity per unit volume at constant strain, is defined as

OU 6) Of] c~ =- 0(3 - a T '

(24)

Thermopiezoelectric smart structures 133

from Eq. (23) follows

Orl _ Oev - ( 2 a )

0 0(3 (3

The relation between thermal, electric and mechanical state variables can now be written by using Eqs. (18) and (20)-(25) in the following form:

Ti j = C~yuk , l - e~i jE~ - ~ j O , (26)

elasticity cony. piezoelectricity thermal stresses

D i = e i j ku j , k -4- ~4ijE j + p i v ~ , (27) direct piezoelectricity permittivity pyroelectricity

o~-: r + p~E~ + oc,/OO , (28) heat of deformation eieetroealorie effect heat capacity

with C~j~l - stiffness tensor, ekij - piezoelectric tensor (relating stress to electric field), {ij - ten- sor of the thermal stress coefficients, x i j - permittivity tensor, pi - vector of the pyroelectric coefficients.

The elimination of the specific heat source r from Eq. (13) using the entropy balance Eq. (19) leads to the inequality

q,~O,~ > o (29) (~ - -

which is satisfied by the Fourier law of heat conduction

q~ = - A~j(~,j = --Aij~),j ,

if the tensor of heat conduction coefficients A~j is positive-semidefinite [21].

(30)

2.3 C o u p l e d f i e l d e q u a t i o n s

The basis for describing the coupled field behaviour is given by the equations of motion (9),

the balance law of angular momentum, the Gauss' law Eq. (3) and the entropy balance Eq. (19):

Tji , j + obi = 042i, T i j = 7~i , Di,~ - O , qi,i = or - O~i?. (31)

With the constitutive equations (26)- (28), the Fourier law of heat conduction Eq. (30) and the vector of the electric field Eq. (3), Eqs. (31) lead to the following coupled system of differ- ential equations [4]:

Cijkl%k,lj -}- eki3~9,kj -- ~ij~),j -w Qbi = oi~i ,

ezkl%k:li -- )gik~) ki + p d ) i ~ 0 ,

(32)

(33)

(34)

For giving a complete description of the coupled three-field problem, the boundary condi- tions on the surface O are required which is supposed to be a union of partial surfaces: 0 = O~ ~5 O r = O~ U OD = O~ U 0r U Oqh where no intersections are expected: O~ ~ OT = O~ N OD = O~ N Oqs N Oqh = 0. In detail, the following boundary conditions occur:

u~ = gi - displacement on O~, T j i n j = [i - traction on OT, (35)

134

~D = @ - surface potential on Oe,

~) = 0 - surf. temp. change on Oo,

Dini = - Q - surface charge on OD,

qini = --G - heat flux across surf. on Oq,,

qini = Oh = h~,( O + Oo - (9oo) - convective heat exchange on 0 ~ ,

where (-) are prescribed values and h. is the convective heat transfer coefficient.

A. G6rnandt and U. Gabbert

(36)

(~7)

(38)

3 Variational and finite element formulation

When introducing the weighting quantities 5ui - virtual displacement, 5p - virtual electric

potential and 5t9 -v i r tua l temperature change, the field equations (31) as well as the boundary conditions Eqs. (35)-(38) can be represented equivalently as follows:

~1~ = f Sui(Tji,j + Qbi - gi~i) dV + f ~ui(ti - Tjinj) dO = O, (39) V Ot

~F e = f 5pDi,i dV - f 5~(0 + Dini) dO = O, (40) V OQ

~Fo = f ~(q~,~ + e ~ , - ~ ) dV - f 6~(0~ + qi~i) dO + f ~O(Oh - q~i) dO = O. (41) V Oq~ Oqh

With the constitutive equations (26), (27) and the gradient equations (3), (30) the Eqs.

(39)-(41) result in the following weak form of the thermoelectromechanical equilibrium,

which gives the basis for the derivation of finite dement formulations:

V V O~

~SF e = - f 5~,i(eijkuj,k -- zijP,j + PiO) dV - f &pQ dO = 0, (43) V OQ

6F~ : f 600(r -Pi@,i + (~cv/(~)O) d V - f 60~r dV V V

+ f 5~,~A~9, ~ d V - f 50G dO + f 5Oh~(O + @o - 0o~)dO = 0. (44) V Oqs Oeh

The field variables ue, ~e, v% within a finite element (subscript e) can be approximated by

means of the shape functions Nu, Ne, No and the unknown nodal degrees of freedom u~, ~ ,

~ via the following relations:

Lie : Nu ({1, {2, ~3) uk, ~oe : N7~ (~1, ~2, ~3) ~Dk, Oe = N0 ({], {~, {3) O~. (45)

For simplification purposes, the symbols

B~ = DuN~, B e = D e N e , Bo = DoN~ (46)

are introduced, where D~, D e, and Do are the differential matrices

-a/az 1

0

0

0 0

O/Oz2 0

0 O/Ox3

O/Oxl 0

O/Ox3 O/Ox2

0 O/Ozl_

D ~ = O/Ox2

0

0/0x3

D~ = Do = I O/Ozl

O/Oz2/ .

0/0X3]

(47)

Thermopiezoelectric smart structures 135

Assuming that 6F~ = ~ 6F~, 6F~ = ~ 6F,~, 6F~ = y~ 6F~, the element energy functional results in

6Fu e = - 5u T f B~CBu dV uk - 6u T f BTerB~ dVqD k + 6u~ f BTr dV ~,~ V ~ V r V ~-

- 6 u ~ f N~t)N~ dV fi~ +6u~ f N ~ b d V +6uk r f NTtdO = 0, (48) V ~ V ~ O[

6 F ; = - f B~eB~dVuk + 6q~ f T B ~ B ~ dV q~ k V ~ V ~

- B~pN~ dVO~ - 5q~ fN~Q dO = 0, (49) vo o~

= N~N~O~" B~ dVu~ - 60~ f NoN~O~ p B~ d V ~ V ~ V ~

f i i V ~ V e V ~

5 ~ f N~O, dO + 5 ~ f T - N ~ h v ( N ~ + @o - @o~) dO = 0. (50) o; 0%

With the abbreviations

NTN ~ ~-TD M e,̀~ = f N~oN~,dV, K e~., = f B~eB~dV, K ~ = f ~ 0 " k ~ DudV,, V ~ V ~ V r

K ~ : f B~CB~dV, K ~ = f BTxB~odV K~e f T T ~ , e = N,0N~O~p B~dV, V ~ V ~ V ~

K ~ f BTeTB~ d r , K,~e f BYpNe dV. Hoo f ~ = e = ~ = N~ op%N~dV, ~tqo V ~ V ~ V ~

(a) K~o = f B ~ ' N o dV, K;~ = f B~'2Bo dV + f NoTh~No dO,

V ~ V ~ 0 ~ ,

f ~ = f N T p b d V + f N ~ t d O , f~o ~ = - f N~ oTQ-dO, w o; o~

f~ fNThv(e~ %) dO+ f T- = - N~qsdO + f N ~ r d V o~ o~ vo

the following semidiscrete system of differential equations for one finite element (superscript e) is obtained:

M r ii ~ ~ e ~ k + Ku~uk + K ~ - Kuo0 k = f~, e e e e K~uk - K~Jf k + K~0k = f~,

Ko~uk - o ~ + Hoo0k + Koo0k = fo.

(52)

(53)

(54)

If the element matrices are assembled into a global matrix, the system equations can be writ- ten as

J ~ [: o :ilil 0 + 0

0 Ko~ - K ~ H~o

0 0 K ~ z f~

(55)

136 A. G6rnandt and U. Gabbert

4 Solution concept

For solving coupled electromechanical problems numerically, a library of powerful finite ele- ments has been developed [10], [11]. If the time dependence of the field variables is neglected and only the stationary case is considered, the set of equations simplifies to

[il - K ~ K ~ = f~ �9 (56)

0 K ~ f~

Now, the complete coupling is split up, and the equations can be solved in two steps:

(i) Calculation of the temperature field:

K ~ = f+. (57)

(ii) Solution of the coupled piezoelectric equations taking into account the thermal strain effect and the pyroelectric effect as force vectors on the right-hand side:

K~ - K ~ f~ - K ~

Extension to the fully coupled three-field problem Eq. (55) requires the unsymmetric matrices to be included in the solution process. Therefore, we decided to solve the three-field problem iteratively. For each time step of the time integration scheme the two sets of equations, the heat conduction equation (59) as well as the piezomechanical equations (60) are solved sepa- rately taking into account the coupling terms as force vectors on the right-hand side of the equations,

H~v4 + K ~ O = f~ - K ~ f i + K , ~ b , (59)

ii

We use the Crank-Nicholson formulae to integrate the heat conduction equation and the Newmark formulae to integrate the piezomechanical equations. For the first time step the initial conditions for temperature, displacement and electric potential are used. For each time step the temperature as well as the displacement and the electric potential are iteratively adapted until convergence is achieved, which means that the fully coupled equations are ful- filled.

5 Example

To illustrate the importance of the coupling effects, the bench mark example presented by T. R. Tauchert [22] is used (see Fig. 1). The five-layer plate with an assumed plane strain state has a length-to-thickness ratio of b/t = 5, each layer is of thickness 0.2t. At the end of the plate the following boundary conditions are assumed: u~i)(x2 = 0, b )= ~(i) ~22 (x2 = 0, b) = 0.

Thermopiezoelectric smart structures 137

piezoelectric

orthotropic, 0 ~ "-~

x2 isotropic

orthotropic, 90 ~

piezoelectric

b Fig. 1. Laminate configuration

The potent ia l at the ends of the piezoelectric layers as well as at the inner surfaces were taken

to be zero, whereas arbi t rary distr ibutions of electric potent ial could occur on the outer sur-

faces. The mater ial propert ies of each individual layer were assumed to be as follows:

| isotropic layer: k7 = E0,/* = z/0, a = c~0, ,~ = A0, 8 = 80, c~ = c~0

| o r thot ropic layer, 0~ E2 = 90. E0, E1 = E3 = 5 �9 E0, G12 = G23 = 4. E0, G13 = 1.5 �9 E0,

/ .12 = /213 = /*/'23 = /'/0, OZ2 = 0 . 0 0 0 2 " O~0, a 1 = O~ 3 = 0.2 - c~0, ~2 = 100 �9 Ao, A1 = A3 -- A0,

8 = 0 .2 .80, c~ = 3 �9 c~,0

| o r thot ropic layer, 90~ E1 = 90 �9 E0, E2 = E3 = 5. E0, G12 = Gla = 4. E0, G2a = 1.5 - E0,

/ '12 = / .13 = /*23 = /*0, OZl = 0 . 0 0 0 2 ' O z 0 , O~ 2 = O~ 3 z 0 . 2 " O~0, h 1 = 1 0 0 ' /~0, /~2 ~ - /~3 z "~0,

<9 = 0.2 �9 80, c~ = 3 �9 cv0

�9 piezoelectric layer: E = E0, /* =/*0, a = a0, A - A0, z l = z2 = • z3 = 10. z0, p = p 0 ,

d31 = d32 - d24 = do, daa = 1.4 �9 do, 8 = 8o, c~, = c~.0.

Fo r our computa t ion the following values were used (see Fig. 1): length b = 50ram,

thickness t = 10mm, Young 's modulus E0 = 2 0 0 0 N m m ~, Poisson's ratio /*0 = 0.25,

thermal expansion a0 = 10e 6K-1 , thermal conductivity A0 = 1 W K - l m i, permitt ivi ty

z0 = le -s F m -1, piezoelectric coefficient do = 2e -1~ mV -1, pyroelectric coefficient

p0 = 0.25e -3 C m - 2 K -1, mass density 8o = 7 600 kg m 3, heat capacity Cv0 = 420 Ws k g - l K -1.

The plate was subjected to a sudden sinusoidal temperature rise 0 = v~0sin(rcy/b), with

~0 = 50 K on the surface z3 = -t/2, while the temperatures at the ends (z2 = 0, b) and on the

surface at3 = t/2 were assumed to be constant at 273.15 K.

F o r investigating the problem each layer was discretized with 26 isoparametr ic plane finite

elements with quadra t ic shape functions. At first, the time dependence of the field variables

was neglected, and Eqs. (57) and (58) were solved. Due to the temperature load, the plate

bends into the z3-direction.The d iagram in Fig. 2 shows the distr ibution of the normalized

displacement (u~ = uatC~olV~olb -2) in the thickness direction at the centre cross section of the

plate. G r a p h a describes the analytic solution by Taucher t [22], neglecting the piezoelectric

effects (direct and converse) and the pyroelectric effect. To compare this result to a numerical

solution in a F E M calculat ion (graph b), the piezoelectric and pyroelectric coupling was also

neglected. The result revealed a very good agreement with the analytic solution. G r a p h c

shows the computed plate bending where also the piezoelectric effects and the pyroelectric

effect were included. The results demonstra te that the piezoelectric effects and the pyroelectric

effect exert a strong influence. Figure 3 shows the difference of the electric potent ial over the

lower piezoelectric layer. Curve a represents the F E M solution including the piezoelectric

effects; graph b indicates the solution when the pyroelectric effect is also considered. This

representat ion shows that it is essential to consider the pyroelectric effect, in part icular , in sen- sor operat ion.

I f the time dependence of the field variables is considered, the coupled Eqs. (59) and (60)

must be solved. Figure 4 shows the temperature at the point [z~ = 0.5, z~ = -0.4] over the

138 A. G 6 r n a n d t and U. G a b b e r t

-0.055

U3*

-0.06

-0.065

-0.07

-0.075 b (uncoupled, numerical)

-0.08

-0.5

�9 c (coupled, numerical)

-0.3 -0.1 0.1 0.3 depth, x3* 0.5

Fig. 2. N o n d i m e n s i o n a l

t ransverse deflect ion

u;[o 5, x~]

250 t t 2 0 0 . . . . . . . . . . . . . . . ! - - i . . . . . . . . . . . . . .

'~176 ..... / ............ .......... / \ ...... 1

q~ IV]

0 0.2 0.4 0.6 0.8 length x2* 1 Fig. 3. Electric potent ia l

[V] at [z~ , -0 .5]

320

| [K]

310

300

290

280

270

0.01

0.2

. . . . . . . . . . . . . i . . . . . . . . _ - . . . . . . ~ . . . . . . o A O [ K ] , l0 -3

-0.2

-0.4

-0.6

. . . . . . . . . . . . . . : : -0.8

0.1 10 lgtime [s] 100

Fig. 4. T e m p e r a t u r e @ and t empera tu re differ- ence A@ at [0 .5 , -0 .4] over the t ime

Thermopiezoeiectric smart structures 139

logarithm of time. The line a describes the uncoupled solution of the temperature field only,

while graph b represents the coupled solution under consideration of the electrocaloric effect.

The effect of heat of deformation is very small in this example so that it is not considered. The

carve c represents the difference between the solutions a and b and shows that the electro-

caloric effect is also relatively small in this example.

6 Conclusion

The paper presents the weak form of the fully coupled thermopiezomechanical field equa-

tions, including linearized constitutive equations, developed on the basis of the balance laws

of mechanics and electrodynamics and the entropy production inequality. The objective was

to review these equations in order to establish a useful concept for a numerical solution

scheme based on the finite element method. The result shows that, upon a finite element dis-

cretization in space, the governing semidiscrete system of differential equations is unsym-

metric. The temperature is coupled with the mechanical displacement and electric potential

only via the first time derivatives. For static cases these equations can be solved separately,

and consequently, also for dynamic cases a separate solution scheme is proposed where the

solutions for each time step are calculated iteratively. An example demonstrates that tempera-

ture may exert a considerable influence on the behaviour of a piezoelectrically controlled

smart structure, which has to be taken into account.

Acknowledgement

This work has been supported by the German Research Foundation. This support is gratefully acknowl- edged.

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Authors' address: A. G6rnandt and U. Gabbert, Institut ffir Mechanik, Universit~it Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany (E-mail: ulrich.gabbert@mb.uni-magdeburg.de)