THEORY OF THE ATTENUATION OF ELASTIC WAVES IN INHOMOGENEOUS SOLIDS

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THEORY OF THE ATTENUATION OF ELASTIC WAVES IN INHOMOGENEOUS SOLIDS

P. Klemens

To cite this version:

P. Klemens. THEORY OF THE ATTENUATION OF ELASTIC WAVES IN INHO- MOGENEOUS SOLIDS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-102-C6-104.

�10.1051/jphyscol:1981632�. �jpa-00221568�

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CoZZoque C6, suppZ6ment au n012, Tome 42, de'cembre 1981 page C6-102

THEORY OF THE ATTENUATION OF E L A S T I C WAVES I N INHOMOGENEOUS S O L I D S

P.G. Klemens

Dept. o f P h y s i c s and I n s t . o f M a t e r i a l s S c i e n c e , ~ n i v . o f C o n n e c t i c u t , S t o r r s , CT 06268, U.S.A.

Abstract.- Temperature variations accompanying elastic waves cause heat conduction, entropy generation and thus attenuation of the waves. In homogeneous solids the temperature variations over half a wave-length and those between different phonon groups cause comparable attenuation. In inhomogeneous solids the temperature gradients are enhanced. The resulting attenu- ation exceeds that due to Rayleigh scattering except at very high frequencies or at low temperatures. Small inclusions and fiber-matrix composites are discussed.

1. Introduction.- In homogeneous single crystals the attenuation of elastic waves arises from the cubic anharmonicities. Kirchhoff first proposed a heat conduction mechanism for gases,") which can also be applied to solids. Attenuation due to the smooihing out of temperature differences between different groups of phonons was treated by

Akhie~er.'~) In both cases do the elastic waves set up temperature differences, and the irreversible heat transfer generates entropy so that elastic energy is converted into heat,

There are two additional attenuation processes in inhomogeneous solids. One is Rayleigh scattering, owing to variations in the local value of the wave velocity. The other is an enhancement of the heat conduction mechanism, because temperature differences are set up be- tween neighboring regions of different thermal expansivity, so that temperature gradients are increased. This was discussed by Zener. ^{( 3 )} It will be seen that the latter mechanism is more important than Rayleigh scattering except at very high frequencies or at low tem- peratures.

2. Homogeneous Solids.- The adiabatic temperature change due to a dilatation A is given by

dTs= -yTA (1)

where y is the ~rcneisen constant. If either A or y varies with position, T will likewise vary and obey

a ~ / a t = ~ v 2 T ₊ a ~ ~ / a t (2) where D is the thermal diffusivity. If y is independent of position,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981632

one c a n w r i t e f o r a wave

A=Aoexp i ( k . g + w t ) and aT=T1 exp i(&.g+wt) ( 3

and from ( 2 )

iwT1= -k DT' 2 - iwyAo ( 4 )

P e r u n i t volume, t h e r a t e of e n e r g y l o s s from t h e wave i s r e - l a t e d t o t h e r a t e o f e n t r o p y p r o d u c t i o n by

-dE/dt=T d S / d t = ( K / T )

1

1

where K = D C i s t h e t h e r m a l c o n d u c t i v i t y , and C i s t h e s p e c i f i c h e a t p e r u n i t volume. The e n e r g y c o n t e n t p e r u n i t volume o f t h e wave i s E=&xA,, 2 where X i s t h e b u l k modulus. Thus one f i n d s f o r t h e r a t e o f e n e r g y l o s s

1 2

I / T = -E- d ~ / d t = y ( ~ / ~ ~ ) u ~ w ~ ( u 2 + w2)-'

where To i s d e f i n e d e q u a l t o X / C , and t h e c h a r a c t e r i s t i c f r e q u e n c y wo i s d e f i n e d a s

w = k D . 2

0

NOW wo depends on t h e wave f r e q u e n c y w , s i n c e k = w / v , . w h e r e v i s t h e wave v e l o c i t y . A t a l l b u t t h e h i g h e s t f r e q u e n c i e s w o < < w , s o t h a t t h e K i r c h h o f f a t t e n u a t i o n becomes

2 2 2

1 / = ~y (T/To)(D/v ~ ) w ⁽⁸⁾

The A k h i e s e r mechanism c a n a l s o be d e s c r i b e d by ( 6 ) , e x c e p t t h a t i n p l a c e o f y 2 one must now u s e

y 2 ,

-2 2 2

1 / = ~y (T/To)(3Dg/v ~ ) w ^{( 9}

While f o r m a l l y 1 / ~ * e x c e e d s 1 /by a f a c t o r 3 , ~ ~ it may w e l l be t h e s m a l l e r o f t h e two, s i n c e y2 must be l e s s t h a n y 2 and s i n c e D i s

g o n l y t h e l a t t i c e component, n o t t h e t o t a l D . A f u r t h e r r e d u c t i o n i n l / ~ p e r h a p s by a f a c t o r 2, r e s u l t s from t h e i n c l u s i o n o f N-proces- ~ , s e s i n t h e d e f i n i t i o n of wo.

3. Inhomogeneous S o l i d s . - Temperature g r a d i e n t s and e n t r o p y g e n e r a - t i o n a r e enhanced i f t h e t h e r m a l e x p a n s i v i t y o r y depend on p o s i t i o n . I f t h e F o u r i e r components o f y ( g ) have p r i r l c i p a l components o f wave- number q and i f t h e i n h o m o g e n e i t i e s a r e o f s m a l l s c a l e s o t h a t q > > k , A c a n be t r e a t e d a s i n d e p e n d e n t o f p o s i t i o n , s o t h a t q 2 r e p l a c e s k 2 i n ( 4 ) and ( 7 ) , and ( 6 ) s t i l l h o l d s w i t h wo r e d e f i n e d . However, i n

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p l a c e o f y 2 one must u s e -2

y - - C viyi 2 - ( C v . y . 1 2

1 1 ( 1 0 )

where vi i s t h e f r a c t i o n a l volume o f t h e component o f Griineisen con- s t a n t yi. I f t h e s m a l l e s t d i a m e t e r o f t h e i n h o m o g e n e i t i e s i s L, wo=a 2 D/L, where D i s t h e s m a l l e r o f t h e t h e r m a l d i f f u s i v i t i e s . 4 . Some Numerical C a s e s . - C o n s i d e r s m a l l c o l l o i d i n c l u s i o n s s u c h a s p r o d u c e d by n e u t r o n i r r a d i a t i o n i n c r y s t a l s , w i t h ~ = 1 x 1 0 - ~ cm. Take T0=25,000 K , T.300 K , and D=vL/3, s o t h a t u 0 = 3 x l 0 l 1 s e c - l . Now

y 2 = n ~ o ~ y 2 , where Ay i s t h e d i f f e r e n c e between y o f t h e i n c l u s i o n a n d m a t r i x , n t h e i n c l u s i o n d e n s i t y , Vo t h e volume o f e a c h i n c l u s i o n . L e t A y = l , t h e n f r o m ( 1 0 ) a n d f r o m ( 6 ) w i t h w < < w o

l / r = 3 n 1 . 0 - ~ ~ n ~ ~ w ~ = 1 . 5 x 1 . 0 - ~ ~ w ~

i n i n v e r s e s e c o n d u n i t s , t h e second v a l u e b e i n g f o r nVo=0.005. T h i s may be compared w i t h R a y l e i g h s c a t t e r i n g by t h e s e i n c l u s i o n s w i t h

Gv/v=$ and v=3x105 cm/sec :-

3 4

l / = ~~ V ~ ( V ~ / V V ~ )w = 2 x 1 0 - ~ ~ n ~ ~ w ~ ( 1 2 ) s o t h a t R a y l e i g h s c a t t e r i n g i s r e l a t i v e l y weak below 6 GHz. A t low t e m p e r a t u r e s a t t e n u a t i o n by h e a t c o n d u c t i o n d e c r e a s e s a s TC.

A s a n o t h e r example c o n s i d e r a c o m p o s i t e o f f i b e r s embedded i n a c o n t i n u o u s m a t r i x , e a c h component o c c u p y i n g h a l f t h e volume, w i t h f i b e r d i a m e t e r 1 x 1 0 ~ ~ cm and D = 1 cm l s e c , s o t h a t 2 s e c - l . With t h e same m a t e r i a l p a r a m e t e r s ~ / T = ~ X ~ O - ~ ~ U ~ S ~ C - ~ d u e t o h e a t con- d u c t i o n , w h i l e f o r R a y l e i g h s c a t t e r i n g 1 / ~ ~ = 8 x 1 0 - 1 9 w 3 . The a t t e n u a - t i o n by enhanced h e a t c o n d u c t i o n i s much s t r o n g e r t h a n i n t r i n s i c a t t e n u a t i o n i n e i t h e r component, and e x c e e d s R a y l e i g h s c a t t e r i n g below a b o u t 1 0 MHz.

T h i s work was done i n c o l l a b o r a t i o n w i t h t h e E n g i n e e r i n g M a t e r i a l s D i v i s i o n o f t h e Naval R e s e a r c h L a b o r a t o r y .

R e f e r e n c e s

( 1 ) G . K i r c h h o f f , P o g g e n d o r f f ' s Ann. 1 3 4 , 177 ( 1 8 6 8 ) . ( 2 ) A. A k h i e s e r , J . Phys. (USSR)

1,

( 3 ) C . Zener " E l a s t i c i t y a n d A n e l a s t i c i t y o f M e t a l s " Univ. o f Chicago P r e s s , C h i c a g o , 1948.