THE SPECTRAL STRUCTURE OF THE PROBLEM OF MAGNETIC ELASTICITY OF THIN P L A T E S

A. L . R a d o v i n s k i i

I z v . AN SSSR. M e k h a n i k a T v e r d o g o T e l a , V o l . 22, No. 1, pp. 164-171, 1987

UDC 539.3

Based on asymptotic analysis, approximate methods of solution of the problem of free oscillators of magnetically elastic plates of a finite size [1,2] are studied. The asymp- totic properties of the spectrum of eigenvalues of the problem are formulated as a func- tion of magnetic field intensity.

! 1. We proceed from the following equations, assuming that oscillations occur as (exp ( ~ w T ) (T is time):

F- ( Q ) ) ,+- ( Q ) ) I-, f- (dcDldrJ ,+- (dcD/d~l)~- 6-[3(i--9) I", fi-(%hpa)-', ~-(2hpa~)'~~~

Ax-w1/E, i-IT

Here k, j = 1, 2 are indexes (summation is performed over recurring indexes; the firs5 equality is equivalent to two equations obtained at k 1 and k = 2); xl, x2, x3 are param- eters of the triorthogonal coordinate system defined In the space V surrounding-the plate, where the median surface S of the plate is coincident with the coordinate surface ra=O. B,(BoI, Bo,,Btr)are values of the vector of magnetic induction on S (.BO is assumed to be known from the solution of the magnetostatic problem and its components to differ from identical ze- ro); u(nt, Q, US) is the vector of displacements of the median surface of the plate; 0 is a potential function (the disturbed component of the magnetic induction b-grad@); the sub- script s indicates that the value of the respective variable is taken on the surface S (at x2 = 0 ) ; (.. .),*- (...)-*@, f-(ba), is the normal component of b in the plate (which, according

J

to C1,21 is assumed to be constant with respect to thickness); A and A s are the three- -

dimensional (in V) and two-dimensional (on S) Laplacians, respectively; L k j are raniliar

operators of the theory of elastic plates; h is the half-thickness of the plate; u,, is the

aaqnetic constant; o , p , E , v are electric conductivity, density, Young's modulus, and Dois- son's ratio of the plate material; X is the frequency parameter; and j,-Jll are arbitrary - factors introduced for convenience (provisionally, they are set eaual to 1).

Equations (1.1)-(1.5) are derived from the equations of [1,2], excluding the tangen- tial components of the electric field induced in the plate and omitting the terms that are beyond the accuracy limits of the Kirchhoff-Love hypotheses. The magnetic permittivity constant is set eaual to 1. In the analysis of an induced rna~netic field in the medium no difference is made between the face surfaces and the median surface of the plate.

The thickness-averaged vectors of the electric current I and the electric Field e In- duced in the plate can be obtained from the pertinent formulas of [1,2;, and -.gritten in terms and with accuracy of (1.1)-(1.5): Q 1987 by Allrrton Press. Inc.

xhere ( i , , i,, ia) a r e u n i t v e c t o r s o f t h e c o o r d i n a t e sys tem on S.

I n s o l v i n g p a r t i c u l a r problems, c o n s t r a i n t s o f t h e f u n c t i o n @ on i n f i n i t y and condi - t i o n s on t h e p l a t e edge r should be added t o ( 1 . 1 ) - ( 1 . 5 ) . These a d d i t i o n a l r e l a t i o n s w i l l be d i s cus sed i n S 4 .

2 . Consider f i r s t t h e spectrum of e i g e n v a l u e s o f t h e problem o f o s c i l l a t i o n s of a n i n f i n i t e p l a t e i n a cons t an t magnet ic f i e l d (BoczO, 8Barlazt-0, i, I-1,2,3) s loped w i t h r e s p e c t t o S .

S o l u t i o n s of ( 1 . 1 ) - ( 1 . 5 ) bounded as lzJl-w w i l l be found i n t h e form (m a r e wave numbers , k= (m,'+ mtz)") : 1 , 2

a,-&' em [ i (m,z ,+m,s) I , F-F' exp [ i (m,+,+m~z,) ]

0-(0,' erp [i(rnlzl+ m,t,)-'kzrl ( i s > + O ) 0 - (@) . - exp [ i (mIt l+m,a)+kzs] (GC-0)

S u b s t i t u t i n g ( 2 . 1 ) i n t o r e l a t i o n s ( 1 . 5 ) , w e o b t a i n e q u a l i t i e s which imply t h a t

(0) ,+-- (0) ,--'IXF', f--'l,kF" exp [ i ( m I z ~ + m t t ~ ) ] ( 2 . 2 ) 1 i

i S u b s t i t u t i o n of u.,, F , and f i n t h e form of (2 .1 ) and ( 2 . 2 ) i n t o (1 .1 ) - (1 .3 ) p roduces

j a system of f o u r homogeneous a l g e b r a i c e q u a t i o n s w i t h c o e f f i c i e n t s c o n s t a n t r e l a t i v e t o ! u j and F', which have a n o n t r i v i a l s o l u t i o n s u b j e c t t o t h e e q u a l i t y i

It i s t h e frequency e q u a t i o n of t h e s even th d e g r e e i n X, which d e f i n e s t h e complex e iqenva lue s w of problem ( 1 . 1 ) - ( 1 . 5 ) .

j An a n a l y s i s of t h e r o o t s of ( 2 . 3 ) i n d i c a t e s t h a t , g e n e r a l l y , t h e f o l l o w i n g app rox i - i mate s o l u t i o n s a r e g o s s i b l e ( J ' i s a number whose v a l u e w i l l be s e t l a t e r ) :

In d e r i v i n g t h e s e e q u a t i o n s , t h e a sympt o t i c s w i th r e s p e c t t o t h e r e l a t i v e h a l f - t h i c k n e s s of t h e p l a t e n = h/R (R i s a l i n e a r s i z e t aken a s a u n i t ) . The v a l u e s of c o e f f i c i e n t s ( 2 . 3 ) a r e expressed a s powers of n :

1 ?/here a and g a r e g iven numbers, and p and r a r e a r b i t r a r y numbers. The numbers a , g, ? ? j and r a r e assumed t o be such t h a t t h e v a r i a b l e s marked by a deg ree s i g n i n t h e r i gh t -hana

F i g . 1

s i d e s of t h e e q u a l i t i e s a r e o f t h e o r d e r o f O(nO). The number g t h e n d e f i n e s t h e asympto- t i c behav io r o f t h e i n t e n s i t y o f a magne t ic f i e l d w i th r e s p e c t t o (2E14)". The number p c o i n c i d e s i n i t s s e n s e w i t h t h e v a r i a b i l i t y i n d i c a t o r [3 ] , and r de t e rmines t h e a s y m p t o t i c s o f t h e f requency parameter A. The numbers p and r ought t o s a t i s f y t h e i n e q u a l i t i e s :

which d e f i n e t h e a p p l i c a b i l i t y o f two-dimensional s h e l l t h e o r y [4] and t h e hypo the se s o f magnet ic e l a s t i c i t y [I]. The last i n e q u a l i t y I s e q u i v a l e n t t o t h e c o n d i t i o n (2h)1p.aoal o f t h e absence o f a s k i n e f f e c t i n t h e p l a t e . I n a d d i t i o n , e ach o f t h e s i x approximate so lu - t i o n s l i s t e d under ( 2 . 4 ) h o l d s , a c c u r a t e t o terms t h a t a r e a s y m p t o t i c a l l y small compared w i th t h o s e r e t a i n e d i n (2 .41 , on ly w i t h i n t h e v a r i a t i o n r e g i o n o f t h e pa r ame te r s r , p, g d e f i n e d by t h e e x p r e s s i o n s

I n e x p r e s s i o n s ( 2 . 4 ) , we assume t h a t j' 0 a t r > -p and j' = 1 at r < -p.

I n t h e th ree-d imens iona l space o f pa r ame te r s ( p , r , g), e x p r e s s i o n s (2 .7 ) and (2.8; de te rmine t h e p l a t e s d e f i n e d by t h e f i r s t e q u a l i t i e s o f ( 2 . 8 ) and l i m i t e d by i n e q u a l i t i e s ( 2 . 7 ) and t h e co r r e spond ing i n e q u a l i t y i n ( 2 . 8 ) .

The s t r u c t u r e o f t h e spectrum f o r a g i v e n i n t e n s i t y o f s t a t i o n a r y magnet ic f i e l d , which by v i r t u e o f ( 2 . 5 ) d e f i n e s t h e v a l u e o f t h e parameter g , can be o b t a i n e d w i t h a s ec - t i o n o f t h e 3 e r t i n e n t p l a n e s by t h e p l a n e o f t h e l e v e l g = c o n s t .

I n F ig . 1, t h e s e c r o s s s e c t i o n s are shown f o r a s t a i n l e s s s t e e l ? l a t e w i t h c h a r a c t e r - i s t i c s i z e s h-10-' m , q-10". I n t h a t c a s e , from ( 2 . 5 ) we have a l l , : , B*=(B,J=5.10z'w t e s l a .

Diagrams ( p , r ) f o r g i v e n c h a r a c t e r i s t i c v a l u e s o f g c an be o b t a i n e d i f , i n F i g . 1, we c o n s i d e r o n l y t h e l i n e s enc lo sed between t h e p o i n t s w i t h i d e n t i c a l s u b s c r i p t s . The l i n e s AiEi r e p r e s e n t f i r s t s o l u t i o n s , BiEi second s o l u t i o n s , KiLi t h i r d , CiFi f o u r t h , 3LBi

f i f t h , and B,D; s i x t h s o l u t i o n s . The v a l u e s o f t h e s u b s c r i p t i a r e d e f i n e d by t h e i n t e n - I L

s i t y of t h e magnet ic f i e l d . - For g < -1.333 (B" < 7 . 1 0 ' ~ t e s l a ) we must se t i = 0, at -1,333<g<-1.183 (7.10-a<B*, t e s l a < 0 . 2 ) - i = 1 a t -1.183<g<-0.583 (0,2<B*, t e s l a < 12 .5 ) - i = 2 at -0,583<g<0,017(125<BL.tesla < 7 9 0 ) - i = 3, a t 0.017 < g (790 t e s l a < B * ) . Note t h a t t h e l a s t two r a n g e s a r e never ach ieved i n p r a c t i c e ; t h e i r a n a l y s i s i s of a p u r e l y t h e o r e t i c a l i n t e r e s t .

3 . A s o l u t i o n of t h e problem of o s c i l l a t i o n of a f i n i t e p l a t e i n a magnet ic fLe ld can be o b t a i n e d approximate ly by u s i n g a n a sympto t i c method deve loped f o r s h e l l o s c i i l a t i o n s i n a vacuum i n [ 4 ] and g e n e r a l i z e d i n [ 5 ] t o t h e o s c i l l a t i o n s of s h e l l s i n a c o m ~ r e s s i b l o l i q u i d .

According t o t h i s method, we p e r f o r m i n ( 1 . 1 ) - ( 1 . 5 ) s u b s t i t u t i o n s of c o e f f i c i e n t s and v a r i a b l e s and a s t r e t c h i n g of t h e s c a l e on t h e b a s i s of e q u a l i t i e s ( 2 . 5 ) (where B.P(r,,z,)=B,,I /rnaxlB,I, q-8-maxlB,1/(2El*)'h) and e g u a l l t i e s

Here p, r, and g have the same meaning as in $2, while c, d, and t establish the re- ciprocal asymptotics of the desired functions;-they will be defined later.

These substitutions take Eqs. (1.1)-(1.5) to a form in which with the values of the ~arameters ( x ) = ( c , d , t. k,, p, r, g ) fixed in some way the reciprocal asympt~ti~s of the terms in the equation are defined by the powers xn of the factors introduced in (1.1)-(1.5): j.-qa,

where xi---2p+e+k,, ~~=-2A-c+k,. XS--g-l-p+t+k,, x,-1-p, y.~-2-4p+d+k,, x , - W d f k : , x,=-g-i-p+t+kz. x8=-a-i+gl-t-p- - r, 733 g+t, X I O = ~ , xll-c .

For the numbers K, formally consistent combinations arc chosen that fulfill the re- 2uirernents of 141; basically, it is the requirement that a limiting process (at r~ + 0 ) be gossible such that it would produce a formally consistent limiting equations system, ob- tained from (1.1)-(1.5) by setting

The number of a variant will be identified by numerals N, n, and q and denoted as 3(N,n,a). The first digit N takes values from 1 to 4 and depends on the choice of the formula (c, a, kl, k 2 ) :

N-I - C-0, d-1, li,-?p, ka-2p-I, r=-p, f i g l .

N-2 - c-I, d-0, k, -2p-I, k,-4p-2, r r l -2p , g<ga

N-S-c-I-min(2p, 2r), d-I-min(2-4p, 2r), k,-k:=-l, r a - i - p , g<g3 (3.2)

N-4 - c-max(0, I ) , d-0, kl-k,-2p-1, r 1 l 2 ( 1 - 2 p ) , g,<g (g+g,)

where 1--1-g+t+p, gr-'/:max(-3+3p, -2a+i+p), and at a minimum one (any) of the last two relations corresponding to N = 2 contains the equality sign.

Expression (3.2) must be supplemented by formulas defining t, gl, g2. Their choice

depends on the relationships of the parameters p and r, and the third digit q in the num- ber of the variant Is chosen accordingly. It has the following values: q = 1 at p+r>a-I. In this case, t--1-a-g+p+r, gl-'12(p-a), g,--'l.(a+l)+p ( 6 1 - 2 p ) , g,-'/*(r-2-a)+Zp (r>l-2p); q = 2 when p f 6 a - L Here t--g, gl='I1(p-I), gz-'12(p-I).

The relations limiting the values of the parameters r and g in (3.2) (the last two for each N) define the portions of the space of parameters ( p , r, g) (generally planes limited in some fashion) in which the respective combinations ( c ,d . kc, k.) are noncontradic- tory.

The transition to limiting systems is done according to rule (3.1), where nl, n2, n3 for a given N are

N-i - n,-4, 5; n,-3, 8, 9, 10; nr-i, 2, 6, 7, 11

N-2 - nt-2, 4. 11: n.96, 7, 8, 9; n8-.I, 3, 5, 10

N-3 - n,-4, 10, il; --I, 2, 5, 6; n.93, 7, 8, 9

N-4- c-4, 5; 118-1, 5 8, 9, 11; nr-3, 6, 7, 10

and the factors j must be assigned the values "2

(0, I ) , re-P

[ (0, I ) , r<l-2p

(1. I ) , --P. (i3.i.)- (i, I ) , -1-2p, ( t o ) , r>-p (I, O ) , r>i-2p

(0, i), r>i-a-p

( I . O), r<l-a-p

This means that in all variants with q = 1 the values (j., j , ) = ( O , I), and in the variants

w l l ~ h q = 2 the values Js = 1. In variants corresponding to N = 3, we will always have ( j 8 3

jg) = (1, l), and the digit q can in that case have the only value q = 2.

The digit n which remains undefined in the variant number is determined depending on the relationship of' the values of the factors j for a given N. These correspondences

2 - are specified by the following expressions: B(1, 1, q) -I,-jIp-0; B ( i . 2, q) - j , - j l o - 1 : B(2, i , q)-1,-1, j7-0; B(2, 2, q) - jI-1,-i; B(Z 3, q ) - is-0, iv-i: B(3, I, 2) - 1:-1,-i, j t - j,-0: B(3, 2, 2) - j,-i*-i, j,-0, jx-0.i; B(3 , 3, 2 ) - - jl-il-i, j,-0, jl- 0. i; B ( 4 1, q) - 11-0, j , - j l ~ - i ; B(4, 5 q) - 11-1, j r = j ~ s - O .

4. Limiting equations derived from (1.1)-(1.5) are:

in the variants B(N,n,l):

TAP-A

A@-0, (@).+--(a).--F, (dcDla~,),+-(a@/&,).-

and in the variants B(N,n,2):

where A indicates the respective limiting form of the right-hand side of (1.3). The cor- I responding limiting forms of (1.1)-(1.2) should be added to the above equations, which caR be written as follows (this also applies to A): I

for the variants B(l,n,q): I

L~~+A'u.-- ia@B*dFld~~ (4.5)

for the variants B(2,n,q): I

L+~cr,--BBataFla~, A-Ad ( B * ~ u ~ / ~ z J

for the variants B(3,n,q):

i~L&+id'ur--?B*,aF/az.

i Y ~ . ' u ~ - j . h ' u ~ - - ~ ~ ~ F / a r , A -0

and for the variants B(4,n,q):

l ~ L ~ + i ~ ~ u ~ - - f i B ~ d F I ~ z .

haul-fiBaidF/dr,

(In (4.5)-(4.12), the summation is done over both subscripts j, k = 1, 2. For variants a(3,nY2), in (4.4) we set j8 = 1.)

Here and henceforth, 13 comparing expressions (4.1)-(4.4: ,:~ith (4.5)-(4.12) it is assumed that the notations N, n, and q retained In numbers of rhe variants can cake any of the values as indicated above.

Each of the reduced limitdng equations systems generally consists of a number of sub-

systems which must be so lved i n a c e r t a i n sequence a c c o r d i n g t o t h e number of t h e v a r i a n t . The sequence i s de f ined by t h e fo l l owing e x p r e s s i o n s :

Re l a t i ons (4 .13 ) should be combined w i t h e x p r e s s i o n s [(42)].-0, (63)-f; k, mrun t h rough t h e va lues k ( 1 , 2 ) and m(1 ,2 ,3) ; t h e exp re s s ion [(a),(b)]+c,d means t h a t t h e v a r i a b l e s c and d can be d e f i n e d by s imul taneous i n t e g r a t i o n of e q u a t i o n s ( a ) and (b), and t h e e x p r e s s i o n (a)--b means t h a t t h e v a r i a b l e b i s d e f i n e d by d i r e c t a c t i o n s from formula ( a ) ; t h e sub- s c r i p t s s o r v a t [.. .] r e f e r s t o t h e r e g i o n ( S o r V ) i n which t h e e q u a t i o n s should be i n - t e g r a t e d . The sequence of a n a l y s i s of t h e equa t i ons i n d i c a t e d i n (4 .13) - (4 .15) must be observed.

When i n t e g r a t i n g on S equa t i ons ob t a ined by t h e l i m i t i n g p roces s from ( 1 . 1 ) - ( 1 . 3 ) , they must be combined a t each po in t o f t h e edge r o f t h e p l a t w i t h two t a n g e n t i a l ( f o r (1.1)) and two normal ( f o r ( 1 . 2 ) ) boundary c o n d i t i o n s ,of t h e s h e l l t h e o r y and p l a t e t h e o r y each [3 ] , and one e l ec t romagne t i c c o n s t r a i n t each; t h e c h o i c e o f t h e l a t t e r i s d i s c u s s e d below. I n i n t e g r a t i n g Laplace equa t i on ( 1 . 4 ) t h e c o n d i t i o n of boundedness on i n f i n i t y should a l s o be f u l f i l l e d i n V.

The e l ec t romagne t i c c o n d i t i o n on r i s chosen so as t o e l i m i n a t e t h e p r i n c i p a l s ingu- l a r i t y of t h e problem of I n t e g r a t i o n of t h e Laplace e q u a t i o n i n V , s o lved e i t h e r subse- quen t ly o r w i t h i n t h e same l i m i t i n g subsystem. The c h o i c e o f t h i s c o n d i t i o n i n t h e gen- e r a l c a se r e q u i r e s a d d i t i o n a l a n a l y s i s . I n two c a s e s it can be formula ted i n t h e i n i t i a l approximation of accuracy accord ing t o t h e c o n d i t i o n s [2 ] : d F / h ( r - 0 ( o r , which I s t h e same, I:lr=O) f o r a p l a t e e i t h e r r e s t r a i n e d o r having a f r e e end i n c o n t a c t w i t h a nonduct ing me- dium, and d F l d n l r - 0 f o r a p l a t e r e s t r a i n e d i n i d e a l l y conduc t ing m a t e r i a l ( h e r e T and n a r e tangent and normal t o r ) . By v i r t u e o f ( 1 . 6 ) , a c c u r a t e t o 0 (ql-'), t h e e q u a l i t i e s (I.n)r-0 ar,d ( e X n ) , = O , w i l l ho ld , r e p r e s e n t i n g t h e absence of a normal c u r r e n t component and of a t a n g e n t i a l component of t h e e l e c t r i c f i e l d on t h e p l a t e edge, r e s p e c t i v e l y .

5 . As i n [4 ,5 ] , i t e r a t i v e p roces se s can be c o n s t r u c t e d f o r s o l v i n g 3roblem of t h e e igenva lues of ( 1 . 1 ) - ( 1 . 5 ) ; they a r e based on a consecu t ive s o l u t i o n o f l i m i t i n g e q u a t i o n s corresponding t o t h e d i f f e r e n t v a r i a n t s , a s de sc r ibed i n $ 4 . Within each i t e r a t i v e ?roc- e s s , a fundamental problem can be i d e n t i f i e d which i s s o l v e d f i r s t and c o n s i s t s i n i n t e - % r a t i o n of c e r t a i n ( fundamenta l ) e q u a t i o n s ; i t should admit of n o n t r i v i a l s o l u t i o n s under homogeneous boundary c o n d i t i o n s . D i sc r epanc i e s i n boundary c o n d i t i o n s a r i s i n g i n t h e so- l u t i ono f t h i s problem can be removed by t h e s o l u t i o n o f a d d i t i o n a l problems.

The v a r i a n t s of l i m i t i n g equa t i ons p r e sen t ed i n $4 make it p o s s i b l e t o c o n s t r u c t a s e r i e s of i t e r a t i o n p roces se s . We w i l l c o n s i d e r on ly t h e fundamental problems, assuming t h a t they can have n o n t r i v i a l s o l u t i o n s i f t h e p e r t i n e n t e q u a t i o n s c o n t a i n t h e frequency 3ararnecer X a s one of t h e i r c o e f f i c i e n t s ( t h i s c o n d i t i o n i s f u l f i l l e d f o r a l l t h e v a r i a n t s ?xcect f o r B ( 2 , 3 , 2 ) ) . The cor responding fundamental problems a r e d e f i n e d b r e x p r e s s i o n s acpear ing f i r s t a f t e r t h e da sh i n ( 4 . 1 3 ) - ( 4 . 1 5 ) . Let u s s p e l l them ou t f o r t h e fo l l owing s i x v a r i a n t s :

A comparison of t h i s r e s u l t w i th $ 2 shows t h a t Fhe domains of v a r i a n t s ( 5 . 1 ) c o i n c i d e wi th t h e domains of t h e r e s p e c t i v e numbers i n ( 2 . 8 ) of approximate s o l u t i o n s ( 2 . 4 ) and of f requency equa t ion ( 2 . 3 ) ; t h e f a c t o r j , , has t h e same meaning as j' i n ( 2 . 4 ) , and t h e f o r -

.LA

mulas ( 2 . 4 ) can be ob t a ined by d i r e c t s u b s t i t u t i o n of ( 2 . 1 ) and ( 2 . 2 ) i n t o co r r e spond ing equa t ions ( 5 . 1 ) . Th i s means, i n p a r t i c u l a r , t h a t f o r equa l r e l a t i v e h a l f - t h i c k n e s s n t h e asympto t ic behav io r s of t h e e igenva lues of t h e problems of o s c i l l a t i o n of a f i n i t e p l a t e I and an i n f i n i t e p l a t e g iven i n ( 5 . 1 ) and ( 2 . 4 ) under t h e same numbers w i l l c o i n c i d e . One can a l s o say t h a t f o r a f i n i t e p l a t e t h e e i g e n v a l u e s w i l l have t h e same p r o p e r t i e s a s i n I I exp re s s ions ( 2 . 4 ) ; namely, f o r o s c i l l a t i o n s of v a r i a n t s 1, 2 , and 4 , t h e a sympto t i c i n - e q u a l i t y R e A ~ I r n A . w i l l t a k e p l a c e , i n d i c a t i n g a predominant o s c i l l a t i o n of t h e co r r e spond ing c h a r a c t e r i s t i c s o l u t i o n s i n t ime , wh i l e f o r v a r i a n t s 3 , 5 , and 6 , I m A W R e A , meaning t h a t a t - t e n u a t i o n P r e v a i l s i n t ime over o s c i l l a t i o n s . ( T h i s assumes t h a t a l l e x a c t v a l u e s of t h e r o o t s of ( 2 . 3 ) a r e complex, and ( 2 . 4 ) a r e t h e i r i n i t i a l asympto t ic app rox ima t ions . ) The a sympto t i c s of t h e e igenva lues of problems ( 5 . 1 ) a r e d e f i n e d by t h e l a s t fo rmula o f ( 2 . 5 ) , where r i s g iven by t h e f i r s t o f t h e co r r e spond ing formulas ( 2 . 8 ) .

Re turn ing t o t h e example of 52, we can n o t e t h a t t h e o s c i l l a t i o n s f o r t h e f i r s t two v a r l a n t s can t a k e p l a c e a l s o a t h i g h e r f r e q u e n c i e s cor responding t o t h e v a l u e s - I < ~ G - 0 . 3 , where t h e s h e l l t heo ry [ 4 ] remains v a l i d . The s o l u t i o n of t h e magnet ic e l a s t i c i t y zroblsrn, however, should be c o n s t r u c t e d hwere w i t h e q u a t i o n s d i f f e r e n t from ( 1 . 1 ) - ( 1 . 5 ) ; t h e l a t t e r a r e i n a p p l i c a b l e a t t h e s e f r e q u e n c i e s , because t h e i r unde r ly ing magnetic e l a s t i c i t y hypo- t h e s e s [l] a r e v i o l a t e d a t ,<a-?.

The fundamental equa t i ons co r r e spond ing t o v a r i a n t s B(1 ,2 ,q) and B(2 ,2 ,q ) , noc s p e l l e d ou t h e r e , a r e r a r e l y used f o r a n a l y s i s ( i n F i g . 1 they a r e r e p r e s e n t e d by t h e p o i n t s A,

J and B

1 , 2 , 3 ) , because t h e i r domain i n t h e space ( p , r , q ) a r e t h e i n t e r s e c t i o n l i n e s of t h e

p l anes d e f i n i n g t h e domains of t h e o t h e r v a r i a n t s . The approximate s o l u t i o n of t h e e igen- va iue problems f o r them, g e n e r a l l y , i s more complex. The fundamental problem of t h e v a r i - a n t B ( 2 , 3 , 2 ) has no e i g e n v a l u e s , and t h e v a r i a n t appea r s i n i t e r a t i v e p r o c e s s e s on ly i n t h e s o l u t i o n s o f t h e a d d i t i o n a l boundary problems.

We s e e t h a t i n a l l t h e v a r i a n t s t h e f a c t o r j 4 = 0 . Th i s means t h a t i n t h e i n i t i a l

equa t i ons ( 1 . 1 ) - ( 1 . 5 ) t h e term 2hf a s compared w i th F can always be o m i t t e d . The r e s u l t - i ng e r r o r e q u a l s ~ l - ~ . Th i s a l s o a p p l i e s t o formulas ( 1 . 6 ) .

Within any sequence ( 4 . 1 3 ) , co r r e spond ing t o t h e v a r i a n t s B ( N , n , l ) , a l l c u a n t i t i e s u 1

and F a r e always d e f i n e d f i r s t by s o l v i n g c e r t a i n equa t i ons on t h e s u r f a c e S; one t h e n c roceeds t o problem ( 4 . 2 ) of d e f i n i n g 9 by i n t e g r a t i n g th ree-d imens iona l Laplace equa t i ons f o r a g iven g r a d i e n t of t h e va lue of t h e f u n c t i o n and f o r a cont inuous normal d e r i v a t i o n on S . Thus, t h e s o l u t i o n of any problem of d e f i n i n g um and F and t h e q u a n t i t i e s expressed

through them, which has t h e p rope r ty p + r > a - l , can be found i n a two-dimensional star;ener.t .:iithout c o n s i d e r i n g 'he f i e l d i n t h e su r round ing medium. ( T h i s e r o p e r f y i s g r a c t i c a l l y sl!\ra;.s observed f o r t h e f i r s t f r e q u e n c i e s of t r a n s v e r s e o s c i l l a t i o n s of t h i n p l a t e s c f 2 f L n i t e c o n d u c t i v i t y ; i t s u f f i c e s t h a t i n e q u a l i t y [p /K) 'hR/ (boha)=- i ) be f u l f i l l e d . ) S o l u t i o n s cor responding t o t h e v a r i a n t s B(N,n,2) f o r d e f i n i n g a l l q u a n t i t i e s um and F i nvo lve an a n a l y s i s - o f th ree-d imens iona l problems.

The fo r ego ing r e s u l t s were ob t a ined under t h e assumption t h a t a s t a t i o n a r y magnetic f l e l d B i s d i r e c t e d a t an ang l e t o t h e s u r f a c e S . I f some of t h e comzonents of B equal z e r3 i d e n t i c a l l y , t h e c l a s s i f i c a t i o n o f t h e c h a r a c t e r i s t i c s o l u t i o n i s changed. Thus, from ( 2 . 3 ) i t fo l l ows t h a t t h e normal magnet ic f i e l d (531 = 3,,2 = 0) and t h e 5angenr; ia l

xagne t i c f:eld (3 0 3 = 9) do not a f f e c t t h e f r e q u e n c i e s of t - ansve r se and t a n g e n t i 2 1 os -

c i l l a t i o n s , r e s p e c t i v e l y . An a d d i t i o n a l s t u d y , which l i e s beyond t h e framework o-'he - , resen t pape r , has i r , d i ca t ed t h a t t h i s Lnfluence i s rnani f t s ted i n t e r n s o f t h e o r d e r o(+-lp),

t h a t were omi t t ed i n t h e t r ans i t ' i on from t h e equa t i ons of [ 1 , 2 ] t o r;he s s u a t i o n s (1.1)-

(1.5); its influence is felt at magnetic fields of a much greater intensity than the oblique

magnetic fields (generally, by fl-l times).

! The author thanks A. L. Goldenveizer for attentive guidahce.

R E F E R E N C E S

1. S. A. Ambartsumyan, G. E. Bagdasaryan, and M. V. Belubekyan, flagnetic Elasticity : of Thin Shells and Plates [in Russian], Nauka, Moscow, 1977.

2. S. A. Ambartsumyan, G. E. Baqdasaryan, and M. V. Belubekyan, "Three-dimensional problem of magnetoelastic oscillations of a plate," PMM, vol. 35, no. 2, pp. 216-228, 1971.

1 3. A. L. Goldenveizer, Theory of Elastic Thin Shells [in Russian], Nauka, Ploscow, 1 1976.

4 . A . L. Goldenveizer, V. B. Lidskii, and P. E. Tovstik, Free Oscillations of Thin I Elastic Shells [in Russian], Nauka, Moscow, 1979.

5. A. L. Radovinskii, "Classification of free vibrations of shells containing a li- / quid," Izv. AN SSSR. NTT*, no. 6, p p . 124-135, 1919. 1 i 25 April 1985 Moscow I

! "4echanics of Solids. Available from Allerton Press, Inc., 150 Yifth Avenue, IJea York, j ;sly 10811; (212) 924-3950.