ATTENUATIONS OF PLANE WAVES IN DEFORMABLE, THERMO-ELECTRIC SOLIDS SUBJECTED TO A PRIMARY MAGNETIC FIELD IN AN ARBITRARY DIRECTION

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ATTENUATIONS OF PLANE WAVES IN DEFORMABLE, THERMO-ELECTRIC SOLIDS SUBJECTED TO A PRIMARY MAGNETIC FIELD

IN AN ARBITRARY DIRECTION

Y. Ersoy

To cite this version:

Y. Ersoy. ATTENUATIONS OF PLANE WAVES IN DEFORMABLE, THERMO-ELECTRIC SOLIDS SUBJECTED TO A PRIMARY MAGNETIC FIELD IN AN ARBITRARY DIRECTION.

Journal de Physique Colloques, 1981, 42 (C5), pp.C5-653-C5-658. �10.1051/jphyscol:19815100�. �jpa-

00220968�

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JOURNAL D E PHYSIQUE

CoZZoque C5, suppZdment au nOIO, Tome 42, oetobre 1981 page C5-653

ATTENUATIONS OF PLANE WAVES IN DEFORMABLE, THERMO-ELECTRIC SOLIDS SUBJECTED TO A PRIMARY MAGNETIC FIELD IN AN ARBITRARY DIRECTION

Y. Ersoy

Department of Engineering Sciences, Middle East XeehnicaZ University, Ankara, !i"urkey

A b s t r a c t .

-

The d i s p e r s i o n r e l a t i o n f o r

electromagneto-thermo-mechanical

( E - F 1 T M ) n e waves i n e l e c t r i c a l l y and t h e r m a l l y conducting, magneto-thermo-

v i s c o e l a s t i c (MTVE) media i s i n v e s t i g a t e d w i t h account f o r t h e e f f e c t o f t h e e x t e r n a l primary magnetic f i e l d . I t i s shown t h a t several modes o f E-MTM waves a r i s e depending upon t h e d i r e c t i o n o f t h e magnetic f i e l d . Among them a

p a r t i c u l a r a t t e n t i o n i s focused on a c e r t a i n mode t h e pase v e l o c i t i e s and t h e a t t e n u a t i o n constants o f which a r e obtained and some l i m i t i n g values are discussed.

1. I n t r o d u c t i o n . - The propagation o f E-MTM waves i n m a t e r i a l s subjected t o an e x t e r - n a l magnetic f i e l d has been the f i e l d o f i n t e r e s t d u r i n g t h e l a s t two decades, e.g.

/l-3/.Because t h e disturbance o f waves i n media i s o f i n v a l u a b l e p r a c t i c a l importance i n seismology, acoustics, transmission l i n e s , and e t c . On t h e o t h e r hand, concerning t h e j u s t i f i c a t i o n o f a c o n s t i t u t i v e t h e o r y which has been developed by comparing w i t h t h e r e s u l t s o f experimental i n v e s t i g a t i o n s , t h e s o l u t i o n s o f t h e s e t o f governing equations must be obtained a t l e a s t approximately. Therefore, i n t h e present i n v e s t i - g a t i o n , a r a t h e r general d i s p e r s i o n r e l a t i o n f o r E-MTM plane waves propagating i n

magneto-thermo-viscoelastic

(MTVE) s o l i d s i s obtained, and some l i m i t i n g values o f t h e phase v e l o c i t i e s and of the a t t e n u a t i o n constants are discussed under c e r t a i n assumptions. Therefore, d e a l i n g w i t h t h e mutual e f f e c t s o f magnetic, s t r a i n and t h e r - mal f i e l d s , t h e f o l l owing i s emphasized by concerning t h e wave phenomena i n i s o t r o p i c s o l i d s : ( i ) The equations presented here a r e t h a t i n the l i n e a r theory o f thermo- v i s c o e l a s t i c i t y ( K e l v i n - V o i g t t y p e ) combined w i t h the l i n e a r i z e d Maxwell equations f o r t h e q u a s i s t a t i c magnetic f i e l d system. However, the pondermotive Lorentz f o r c e and t h e a c t u a l s t r e s s tensor f o r t h e s o f t ferromagnetic m a t e r i a l s are taken i n t o account. ( i i) Due t o t h e c o n d u c t i v i t y , t h e t h e r m o - e l e c t r i c e f f e c t s a r e a l s o consid- ered w h i l e t h e mechanical p a r t o f t h e present s t a t e s t r e s s tensor i s n o t determined by the present s t r a i n alone, i . e . a s h o r t memory i s included s i n c e viscous deforma- t i o n s a r e observed i n magnetic m a t e r i a l s a t temperatures t h a t can be a p p r i c i a b l y below t h e i r me1 t i n g p o i n t s /4,p.52/. Thus we hope t h a t a considerable i n s i g h t i n t h e r o l e played b y MTVE coupling, i n t e r m e d i a t e e s t i m a t e o f t h e damping because o f t h e i n t e r a c t i n g f i e l d s , and u t i l i t y i n assessing t h e accuracy o f an approximate theory a r e achieved.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19815100

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C5-654 JOURNAL DE PHYSIQUE

2 . L i n e a r i z e d Governing Equations o f PTVE and D i s p e r s i o n R e l a t i o n : - The f i e l d e q u a t i o n s p r e s e n t e d h e r e a r e t h e s p e c i a l case o f t h e e q u a t i o n s o f m o t i o n and o f c o n s t i t u t i v e e q u a t i o n s o b t a i n e d i n / 5 / . More s p e c i f i c a l l y , t h e mentioned e q u a t i o n s a r e l i n e a r i z e d under t h e assumptions t h a t t h e m a t e r i a l i s s u b j e c t e d t o a u n i f o r m p r i m a r y magnetic f i e l d (PMF)

5

and undergoes i n f i n i t e s i m a l d e f o r m a t i o n s . Thus, e l i m i n a t i n g t h e

p e r t u r b e d e l e c t r i c f i e l d

e

and r e a r r a n g i n g t h e terms, t h e f o l l o w i n g e q u a t i o n s f o r t h e c o n s i d e r e d m a t e r i a l a r e o b t a i n e d /6/

where

h, e

a r e t h e f l u c t u a t i n g m a g n e t i c and temperature f i e l d s ,

u

t h e displacement, p

t h e d e n s i t y , 2 t h e magnetic s u s c e p t i b i l i t y , i? t h e e l e c t r i c a l c o n d u c t i v i t y ,

2

t h e thermal c o n d u c t i v i t y , and Ci t h e e l a s t i c L6ma c o n s t a n t s ,

5

and t h e v i s c o u s Lime c o n s t a n t s ,

i

t h e t h e r m a l c o n s t a n t , & and I? t h e c o n s t a n t s f o r t h e r m o e l a s t i c and t h e r m o e l e c t r i c e f f e c t s , uo t h e p e r m e a b i l i t y o f vacuum, cijk t h e a l t e r n a t i n g t e n s o r , at t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o time, and t h e comma i n t h e s u b s c r i p t s denotes t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t h e s p a t i a l c o o r d i n a t e s xi.

S o l u t i o r s t o t h i s c o u p l e d systems o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e sought i n t h e f o r m o f s t e a d y harmonic Naves moving i n t h e ? - d i r e c t i o n . I n t h i s case,

h,

and

e

must be f u n c t i o n s o f 8 and t i n t h e f o r m

I hk, uk,

e

) = { h i , u;, O* 1 exp

I:

i ( k v x - w t )

1,

( 2

where

h*,

!* and E* a r e c o n s t a n t complex amp1 i t u d e s , and

k

i s t h e wave v e c t o r , w t h e a n g u l a r f r e q u e n c y and

i = ~ r l .

I n v i e w o f ( 1 ) and ( 2 ) , seven a l g e b r a i c e q u a t i o n s o f t h e f o r m

a r e o b t a i n e d , where A& i s t h e 7 x 7 c o e f f i c i e n t m a t r i x Y: i s t h e column v e c t o r d e f i n e d b y

Thus t h e e q u a t i o n s f o r MTVE a r e c o n s i d e r a b l y s i m p l i f i e d because o f t h e assumptions f o r t h e p l a n e waves, and ( 3 ) i m p l i e s t h e c o n d i t i o n

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d e t

I(

^A_aB

(1

^s

1

A($

1

⁼⁰

f o r t h e p r o p a g a t i o n o f t h e wave. The c o n d i t i o n ( 5 ) g i v e s , i n g e n e r a l , a seventh o r d e r p o l y n o m i a l e q u a t i o n i n k 2 , t h e r e f o r e , seven modes o f waves f o r an a r b i t r a r y d i r e c t i o n o f t h e

PMF

may a r i s e .

Now, r e l a t i v e l y g e n e r a l e q u a t i o n ( 5 ) may be s i m p l i f i e d i n t h e case when t h e planes o f c o n s t a n t phase a r e a l s o t h e p l a n e s o f c o n s t a n t a m p l i t u d e , and t h e PMF has on1 y two components, i .e.

,

where Ho i s t h e magnitude o f

8,

and 4 i s t h e a n g l e between

b

and

8 .

A f t e r some l e n g t h y e l e m e n t a r y row o p e r a t i o n s , t h e m a t r i x

1)

^A&

11

i s p u t i n t o t h e upper t r i a n g u - l a r f o r m so t h a t ( 5 ) can b e deduced e a s i l y . I t i s i n t e r e s t i n g t o n o t i c e t h a t ( 5 ) l e a d s t o h; =O s i n c e

9l

f 0 f o r t h e wave. Thus has o n l y t r a n s v e r s e components, and t h e number o f t h e independent e i g e n v e c t o r s s a t i s f y i n g ( 3 ) a r e a t most s i x . Then a l l t h e waves o f s m a l l a m p l i t u d e s may be c o n s i d e r e d as a l i n e a r c o m b i n a t i o n o f such monochromatic waves.

3. Phase V e l o c i t i e s and A t t e n u a t i o n Constants.- To c a r y on t h e f u r t h e r c a l c u l a t i o n s , ( 5 ) i s r e a r r a n g e d i n more e l e g a n t f o r m by i n t r o d u c i n g some d i m e n s i o n l e s s q u a n t i t i e s . L e t 5 and K be, r e s p e c t i v e l y , t h e d i m e n s i o n l e s s wave number and t h e frequency d e f i n e d b y

where wo i s t h e c h a r a c t e r i s t i c frequency, and c o i s t h e speed o f l i g h t i n magnetiz- a b l e m a t e r i a l w i t h K and t h e e l e c t r i c s u s c e p t i b i l i t y

is

zero.

W i t h t h e use o f ( 6 ) and ( 7 ) , and i n t r o d u c i n g some d i m e n s i o n l e s s q u a n t i t i e s , ( 5 ) i s expressed i n t h e f o r m

where t h e f i r s t f a c t o r i s g i v e n by

A f t e r l e n g t h y c a l c u l a t i o n s , and h3 can be expressed i n t h e same f a s h i o n . The q u a n t i t i e s i n ( 9 ) a r e d e f i n e d i n terms o f t h e d i m e n s i o n l e s s m a t e r i a l c o n s t a n t s v ( a = 1,2,. ..) and parameters qa ( a = 1,2 ,...) w h i c h a r e r e l a t e d by t h e PMF. They a r e

a g i v e n as

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JOURNAL DE PHYSIQUE

where

and t h e s u b s c r i p t s M and S stand f o r t h e EM- and mechanical S-waves r e s p e c t i v e l y . Thus, concerning o n l y the f a c t o r dl i n (8). ( 9 ) i m p l i e s t h a t t h e r e a r i s e the coupled and m o d i f i e d mechanical S- and EM-waves because o f t h e PMF, the phase veloc- i t i e s and t h e a t t e n u a t i o n constants o f which a r e determined by s o l v i n g t h e b i q u a d r a t - i c equation

where

a. = r 2 ( 1 + v T K ~ ) ; a2 = [I

-

vl(ql + r12)] + i ( v 2 a o + v l ~ 2 ⁺v1 ⁺n2)

,

I t f o l l o w s from (12) t h a t 6 has t h e complex r o o t s

;cl

^and

^;c2

from which we can e a s i l y determine the phase v e l o c i t y V(H) and t h e a t t e n u a t i o n constant ( r a t e of damping) ~ ( 0 ) . Upon using t h e phase v e l o c i t y V(0) and the a t t e n u a t i o n c o n s t a n t a(0) f o r the S-wave i n t h e m a t e r i a l n o t subjected t o t h e PMF, i.e.,

i n the r e s u l t i n g equations, we o b t a i n

"(H) 1 ( K + 1 ) (H)

= - . ; - - (1+K3)K4

V(O) K1 1 +K2 ~ ( 0 ) - K1 ( K ~ - 7 1 % f o r the m o d i f i e d S-wave; and

f o r the m o d i f i e d EM-wave, where

K O ,

K1,

K2,

K3 and K4 are t h e q u a n t i t i e s depending upon the m a t e r i a l constants (e.g. 5 , 2, v l , ...), t h e domain o f t h e magnetic

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parameters given by (10) and upon the c h a r a c t e r i s t i c frequency of the wave. For

2

394 example,

K =

l+v

~ 2 .

0

1 I t i s a t once apparent from (15) and (16) t h a t the i l l u s t r a t i o n of

V1

,*(H) and

a

(H) .would be a heavy task unless some of the parameters a r e specified, and then some numerical computations are done o r some ideal ized materials a r e concerned. 1,2 Before considering some approximate values of

V

(Hj and

a1

,2(H), i t i s i n t e r e s t i n g to note t h a t the modification of the waves disappear i f the angle 192

I$ =

( n - l ) n / 2 , n

=

1,2,. . . ( i .e., t h e PMF i s e i t h e r longitudinal or transverse) while the coupling do a r i s e . However, f o r the nonvanishing magnetic f i e l d and s u i t a b l e material constants, the waves a r e , i n general, coupled and they consist of predominantly.

mechanical, EM- o r thermal- waves. Thus the r a t e of dampings of these waves deviate from those of the uncoupled waves, and the modified waves continuously i n t e r a c t and exchange energy.

Now, f o r a c l e a r idea about the frequency dependence and the e f f e c t of the PMF on

V1

,2(H) and

a1

,2(H), (15) and (16) can be simplified reasonably f o r some m a t e r i a l s Of a l l many possible values of material constants and the magnitude of the PMF, two types of waves a r e , however, of p a r t i c u l a r importance: those a r e sound waves

(SW)

and ultrasound waves (USW). Depending upon the c h a r a c t e r i s t i c frequencies of

SW

and USW and the relaxation times, i f the parameters in (10) and (11) a r e computed, then

Lim

V1

,2(H)

=

1 Lima ( H ) = O ,

K - ~ O

( l - s , ) % i ( l + c l ) % '

r4-1

ly2

Lim V, , 2 ( ~ )

=

L2 Lim

a

( H )

-t K-

( 1 - s 2 ) r i ( 1 + ~ ~ ) 4 '

^K-

a r e obtained, where

From ( 1 7 ) , i t can be concluded, s t r i c t l y speaking, t h a t the modified waves propagate

without damping i f the dimensionless frequency

^K

approches zero, and they may

s i g n i f i c a n t l y attenuate a t high frequencies. While the e f f e c t of the PMF on

V1

,2(H)

i s , in the low scale frequencies, s i g n i f i c a n t , i t a l s o becomes disappear in the high

s c a l e frequencies. Therefore, t h e modified waves a t the low frequencies s u f f e r t h e

applied magnetic f i e l d more than the waves a t the high frequencies.

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C5-658 JOURNAL DE PHYSIQUE

On t h e o t h e r hand, f o r t h e a p p r o p r i a t e m a t e r i a l constants and very small ( o r l a r g e ) frequencies i f the terms i n v o l v i n g the f a c t o r w 2 ( o r 5 ' ' ) i n ( 1 2 ) i s neglected compared w i t h t h e others, t h e r o o t s o f the r e s u l t i n g equation a r e then computed e a s i l y . The computation i m p l i e s t h a t the wave i s a predominatly S (EM)-wave a f f e c t e d by t h e PMF i f the frequencies a r e v e r y small ( l a r g e ) . Furhermore i f t h e r e i s no e x t e r n a l l y a p p l i e d magnetic f i e l d , o r i f t h e magnetic f i e l d i s s u f f i c i e n t l y weak such t h a t n ² n1

=

n2 2 0, (15) and (16) become r e s p e c t i v e l y

agreeing, as would be expected, w i t h t h e values occur i n t h e uncoupled waves. Here t h e s u b s c r i p t M stands f o r the uncoupled EM-wave. I n p a r t i c u l a r , i f t h e m a t e r i a l i s an e l e c t r i c a l l y i n s u l a t o r , then t h e e f f e c t o f the PMF diminishes. The cases concern- i n g t h e o t h e r modes and some i d e a l i z e d m a t e r i a l s can be dis,cussed s i m i l a r l y .

References /1/ Knopoff, J., J. Geophys. Res.

60,

441 (1955).

/2/ Dunkin, J.N. and Eringen,A.C., I n t . J. Engng. S c i .

1,

461 (1963).

/3/ H u t t e r , K., I n t . J. Engng, Sci.

13,

1067 (1975).

/ 4 / Vonsowskii, S.V., Magnetism, Vo1.2, John Wiley and Sons, Inc., New York (1974).

/5/ Ersoy, Y., METU J. Pure and Appl. Sci.

12,

49 (1979).

/6/ Ersoy, Y., I n t . J. Engng. S c i .

19,

91 (1981).