Acoustic Wave Propagation in Materials with Inclusions or Voids

Chapter 12

Acoustic Wave Propagation in Materials with Inclusions or Voids W. M. Madigosky and K. P. Scharnhorst Naval Surface Warfare Center, Silver Spring, MD 20903-5000

The acoustic properties of homogeneous, isotropic viscoelastic materials are modified significantly in the presence of randomly distributed voids or viscoelastic inclusions. Given a frequency-dependent effective medium theory, effective wave speeds and attenuation properties of the resulting inhomogeneous materials may be calculated by introducing complex valued material parameters for the constituent materials. The resulting effective dynamic parameters are found to reduce to the quasi-static ones originally derived by Kerner, in the long wavelength, low volume concentration limit. The data presented here suggest that the Kerner equations adequately predict the properties of materials with microscopic voids even at moderately high volume concentrations. With increasing volume concentrations and decreasing wavelengths, theoretical results generally diverge. However, in the resonance region of the inclusions, considerable qualitative insight into the dynamic response of these materials may be obtained by studying a certain simple, closed form expression for the effective wave vector. Spherical inclusions are considered throughout this paper.

Many problems in ultrasonic visualization, nondestructive evaluation, materials design, geophysics, medical physics and underwater acoustics involve wave propagation in inhomogeneous media containing bubbles and particulate matter. A knowledge of the effect of voids or inclusions on the attenuation and velocity of sound waves is necessary in order to properly model the often complex, multilayered systems. This chapter not subject to U.S. copyright Published 1990 American Chemical Society

230

SOUND AND VIBRATION DAMPING WITH POLYMERS

Two f a c t o r s w i l l determine the a c c u r a c y of the m o d e l i n g . The f i r s t i s the a c c u r a c y w i t h which the dynamic mechanical p r o p e r t i e s of the c o n s t i t u e n t m a t e r i a l s i s known, and the second i s the degree to which the e f f e c t i v e modulus theory a c t u a l l y models the p r o p e r t i e s of the inhomogeneous m a t e r i a l . The complex shear modulus i s by f a r the most important dynamic p r o p e r t y to i n f l u e n c e the a c o u s t i c p r o p e r t i e s of inhomogeneous materials. The a c c u r a t e d e t e r m i n a t i o n of the dynamic shear p r o p e r t i e s i s not an easy task as they are o f t e n very temperature and f r e q u e n c y dependent. In a d d i t i o n , the dynamic p r o p e r t i e s are d i f f i c u l t ( i f not i m p o s s i b l e ) to measure at the f r e q u e n c i e s of i n t e r e s t . Conseq u e n t l y , s h i f t i n g , u s i n g a time-temperature p r i n c i p l e such as t h a t of the WLF s h i f t (1) i s employed to o b t a i n data over a f r e q u e n c y range l a r g e r than that used i n the a c t u a l experiment. Recently, s e v e r a l methods have emerged which g r e a t l y improve the r e l i a b i l i t y of the dynamic data and o f f e r a much wider a c t u a l measurement range (2-4). The second problem f a c i n g the r e s e a r c h e r , and the one which i s the s u b j e c t of t h i s paper, i s to determine which, i f any, e f f e c t i v e modulus theory a c c u r a t e l y p r e d i c t s the a c o u s t i c wave v e l o c i t y and a t t e n u a t i o n i n a m i c r o s c o p i c a l l y or m a c r o s c o p i c a l l y inhomogeneous material. The approaches are d i v i d e d i n t o those which do not take f r e quency i n t o account (except f o r r e l a x a t i o n phenomena) and t h e r e f o r e are " s t a t i c " t h e o r i e s and those which take frequency i n t o account, g e n e r a l l y by a s c a t t e r i n g approach. The l a t t e r , "dynamic" t h e o r i e s , when reduced to the low frequency or q u a s i - s t a t i c l i m i t , u s u a l l y compare f a v o r a b l y w i t h the s t a t i c t h e o r i e s . Some approaches take m u l t i p l e s c a t t e r i n g i n t o account and cannot be s o l v e d i n c l o s e d form. These r e q u i r e e l a b o r a t e computer number c r u n c h i n g t e c h n i q u e s (5). F i n a l l y , i t i s important to mention the b a s i c l i m i t a t i o n of a l l of the t h e o r i e s w i t h regard to the c o n c e n t r a t i o n of i n c l u s i o n s . Except f o r the l a r g e number crunching approach, which r e q u i r e s a knowledge of the p a i r c o r r e l a t i o n f u n c t i o n , a l l t h e o r i e s are e s s e n t i a l l y l i m i t e d to low or d i l u t e c o n c e n t r a t i o n s of i n c l u s i o n s . The exact l i m i t i s not known, but i t would appear t h a t something on the o r d e r of 10 percent i s r e a s o n a b l e . Most of the t h e o r i e s u s i n g d i f f e r e n t approaches agree up to that c o n c e n t r a t i o n . In p r a c t i c e , however, the m a t e r i a l d e s i g n e r uses these t h e o r i e s up to much h i g h e r concentrations. In what f o l l o w s , we s h a l l attempt to g i v e the reader a sense of the a v a i l a b l e t h e o r i e s and approaches, and the assumptions and l i m i t a t i o n s of each. We w i l l a l s o compare the d i f f e r e n t t h e o r i e s w i t h some e x p e r i m e n t a l d a t a . Spherical Microscopic

Inclusions

S t a t i c Theories. The s i m p l e s t e f f e c t i v e modulus theory i s t h a t which assumes the components behave as an i d e a l l i q u i d mixture of one substance i n another ( 6 ) . In t h i s case, both the d e n s i t y and the c o m p r e s s i b i l i t y are a d d i t i v e p r o p e r t i e s of the c o r r e s p o n d i n g q u a n t i t i e s of the two m a t e r i a l s depending upon the p r o p o r t i o n a l amount of each substance i n the m i x t u r e . Thus,

12. MADIGOSKY AND SCHARNHORST

231

Acoustic Wave Propagation

p = (l-4>)p + 4>

(1)

Pl

and fUa-c^P +

fcPj

(2)

w h e r e p a n d p a r e t h e d e n s i t y a n d c o m p r e s s i b i l i t y o f t h e m e d i u m , p^ a n d P i a r e t h e d e n s i t y a n d c o m p r e s s i b i l i t y o f t h e i n c l u s i o n s , a n d 4> i s the volume f r a c t i o n of i n c l u s i o n s . The t i l d e i n d i c a t e s effective properties. Hence, the e f f e c t i v e c o m p r e s s i o n a l modulus, M , i s g i v e n by

2

M =p =i C

(3)

P

T h i s a p p r o a c h c o r r e c t l y p r e d i c t s the o b s e r v e d e f f e c t i v e sound v e l o c i t y , c , i n the case of metal i n c l u s i o n s i n a soft v i s c o e l a s t i c medium (T_), b u t c a n n o t be a p p l i e d when t h e medium h a s a l a r g e s h e a r modulus. F o r a f l u i d medium o r a s o f t r u b b e r , t h e e f f e c t i v e v e l o c i t y i s

(3

- j =(1+A4>)(1-B4>)

(4)

where

P-P, P

p.-p A= P The v e l o c i t y h a s

(5)

andB=

a minimum ( a s s u m i n g

A , B>0)

at (6)

. =(A-B)/(2AB) nun

For

solids

i n a soft

rubber

the

loss

(see

Equations

a x = ^ 8 3

M

35-39)

is

(7) v

and i s s m a l l . Here a i s the a t t e n u a t i o n c o e f f i c i e n t , A the w a v e l e n g t h , p t h e s h e a r m o d u l u s , and 8 t h e s h e a r l o s s f a c t o r . Major e f f e c t s which g r e a t l y modify the a c o u s t i c p r o p e r t i e s of m a t e r i a l s a r e due t o i n c l u s i o n s w h i c h a r e i n f a c t v o i d s . This case w i l l now be c o n s i d e r e d . One o f t h e most s u c c e s s f u l t h e o r i e s i s t h a t o f K e r n e r (8). K e r n e r employs a t h r e e phase model c o n s i s t i n g o f an a v e r a g e s i z e s p h e r i c a l i n c l u s i o n s u r r o u n d e d by a s h e l l o f t h e h o s t m a t e r i a l a n d i m b e d d e d i n t h e e q u i v a l e n t homogeneous m e d i u m . The i n c l u s i o n s a r e d i s t r i b u t e d r a n d o m l y a n d t h e r e i s no i n t e r a c t i o n b e t w e e n t h e m . Under t h e e f f e c t o f u n i f o r m h y d r o s t a t i c p r e s s u r e and u n i f o r m t e n s i o n , t h e e f f e c t i v e b u l k m o d u l u s , K = 1 / p , and e f f e c t i v e s h e a r modulus, p, are c a l c u l a t e d . The r e s u l t i s p

232

SOUND AND VIBRATION DAMPING WITH POLYMERS (1-4>)K ^ K=

^

t

^K,

3K + 4u

3K.+4u • * i

(1-4))

y

3K + 4p

(8)

3K.+4u

and

1-4)

(7-5o)u + (8-10o)u

15(l-o)

1

£=p —

V ~

+

(7-5o)u + (8-10o)p

+ 1

(9) l-4>

15(l-o)

where K and p a r e the bulk, and shear moduli of the host medium, and the s u b s c r i p t i r e f e r s t o the i t h i n c l u s i o n s . A l s o , o = (3K-2u)/ (6K+2u) i s P o i s s o n ' s r a t i o . The e f f e c t i v e compressional modulus i s g i v e n by M = K + 4p/3

(10)

When the i n c l u s i o n s a r e t h i n - w a l l e d , hollow spheres, such as the g l a s s m i c r o s p h e r e s used i n a s y n t a c t i c foam, then the moduli of t h e i n c l u s i o n s a r e g i v e n by (9) 2E t K =

g

(11)

3 ( l - o )R and

P l

2(l+5o)R

g

Here Eg and Og a r e Young's modulus and P o i s s o n ' s r a t i o f o r the s h e l l m a t e r i a l , and t and R a r e the w a l l t h i c k n e s s and o u t e r r a d i u s of t h e sphere. When the i n c l u s i o n s a r e v o i d s , E q u a t i o n 10 becomes 4p ~ M=

K(l-4>) 3 — + 3K4> 1 +15(l-o)4)/l(7-5o)(l-4>)]

, (13) x

+

Other i n v e s t i g a t o r s (10) have o b t a i n e d r e s u l t s s i m i l a r t o those o f Kerner. The r e s u l t s o f Dewey (_11) which a r e v a l i d f o r a d i l u t e s o l u t i o n agree w i t h the Kerner e q u a t i o n i n the d i l u t e s o l u t i o n limit. C h r i s t e n s e n (1_2) reviews and r e d e r i v e s the e f f e c t i v e modulus calculations f o r spherical inclusions. The t h r e e models which a r e


233


p r e s e n t e d a r e the d i l u t e s u s p e n s i o n , n o n - d i l u t e composite s p h e r e s , and t h r e e phase models. H i s d i l u t e suspension model agrees w i t h the work of Dewey and the d i l u t e suspension l i m i t of the Kerner equation. H i s composite model i s i d e n t i c a l to that o f Hashin ( 1 3 ) . C h r i s t e n s e n s treatment of the three phase model d i f f e r s somewhat from that of p r e v i o u s authors (8.10). H i s r e s u l t s d i f f e r o n l y i n the shear modulus from those of Kerner, although under d i l u t e s u s p e n s i o n c o n d i t i o n s the two agree. The Kerner e q u a t i o n , a t h r e e phase model, i s a p p l i c a b l e t o more than one type of i n c l u s i o n . Honig (14,15) has extended the Hashin composite spheres model to i n c l u d e more than one i n c l u s i o n t y p e . S t a r t i n g w i t h a dynamic theory and going to the q u a s i - s t a t i c l i m i t , Chaban (16) o b t a i n s f o r e l a s t i c i n c l u s i o n s i n an e l a s t i c material 1

M=

it?H 2

U4)

2 u

10 / i + — l - — 3 u

1 +3 3Aj+ 2pj+ 4p where, A = K-2p/3 and

E=

(15) u

i

3A + 8u

P

P

9A + 14p + P

2

For

a voided material ~

A+2u

M=

, (16) %

2 3(A+ -p)4> 3 H

1+ 4u E q u a t i o n s 13 and 16 y i e l d

20u + 9A-fl4p

the same r e s u l t s when 4>< 10 p e r c e n t .

Dynamic T h e o r i e s . Dynamic t h e o r i e s take i n t o account the s c a t t e r i n g of a c o u s t i c waves from i n d i v i d u a l i n c l u s i o n s and g e n e r a l l y i n c l u d e c o n t r i b u t i o n s from a t l e a s t the monopole, d i p o l e , and quadrupole resonance terms. The s i m p l e r t h e o r i e s model o n l y s p h e r i c a l i n c l u s i o n s i n a d i l u t e s o l u t i o n and thus do not c o n s i d e r m u l t i p l e scattering. To o b t a i n u s e f u l a l g e b r a i c e x p r e s s i o n s from the t h e o r i e s , the low c o n c e n t r a t i o n and the low frequency l i m i t i s u s u a l l y taken. In t h i s l i m i t , the v a r i o u s t h e o r i e s may be r e a d i l y compared. The t h e o r y of Chaban (16) i n v o l v e s a s e l f - c o n s i s t e n t f i e l d t e c h n i q u e f o r randomly d i s t r i b u t e d i n c l u s i o n s a t low c o n c e n t r a t i o n s . The f i e l d at each i n c l u s i o n i s equal to the sum o f the i n c i d e n t f i e l d and the f i e l d s s c a t t e r e d by a l l o t h e r i n c l u s i o n s minus i t s own self field. Thus, the s e l f c o n s i s t e n t approach attempts to improve

234


the modeling of the s c a t t e r i n g process but s t i l l does not take i n t o account m u l t i p l e s c a t t e r i n g . Chaban has c o n s i d e r e d the case of resonant v o i d s i n a low shear modulus m a t e r i a l , as i n the case of resonant a i r bubbles i n water. In t h i s case, the d i p o l e , quadrupole, and h i g h e r o r d e r terms may be dropped compared to the monopole term. The e f f e c t i v e modulus when X » p i s g i v e n by

M =

(17) 1 + 3X/4p(l- ( —

)

where

(18)

a

p

i s the resonance frequency (17,18) of a c a v i t y of r a d i u s a, and q i s a l o s s term a s s o c i a t e d w i t h the damping of c a v i t y v i b r a t i o n s . For an a i r c a v i t y i n water t h i s w i l l be a sum of v i s c o u s , t h e r m a l , and r a d i a t i o n terms whereas f o r a c a v i t y i n rubber i t w i l l be e s s e n t i a l l y the l o s s a s s o c i a t e d w i t h the damping of shear waves, i . e . , the shear l o s s f a c t o r . Kligman and Madigosky (L9) extended the work of Chaban to the case of a s o l i d medium. Based on t h i s , i t can be shown t h a t when o n l y the lowest o r d e r terms i n frequency of the monopole and quadr u p o l e c o n t r i b u t i o n s are r e t a i n e d (the d i p o l e c o n t r i b u t i o n appears i n the e f f e c t i v e d e n s i t y ) , the e f f e c t i v e modulus i s 2

[1/

6

\

1 /

2

z

(ka)

(19)

M = (X+2u)/ 1 + 4> 4

X

2

-P+X+-p1

3

1

3

1

2

o

(kar

—

3

where k = a>/c = a>/V(X + 2p)/p and the r a d i a t i o n damping term i n the denominator has been o m i t t e d . T h i s has a resonance f r e q u e n c y a t

4 -P + A

1 +

2 -p

1/2

(20)

1

c (x)

—

°

—

a

a I X V3

2

1

3

^

and f o r s o l i d i n c l u s i o n s i s g e n e r a l l y h i g h e r i n frequency than the d i p o l e resonance. For v o i d s , the monopole i s the lowest frequency resonance and E q u a t i o n 19 reduces to


2 A+ - u -

M= (A + 2p)/

235

Acoustic Wave Propagation (ka,:

(H)

3

l+34>

(21)

4u-A(kar T h i s resonance i s a t 1/2

(22)

- - ( - pv ( 0

A

a Vp V

U2

(23)

k a = 2(pA)

which i s e s s e n t i a l l y the monopole r e s u l t of Meyer, Bremdel and Tamm (17) when p « A , but otherwise p r e d i c t s a h i g h e r resonance f r e q u e n c y . S i m i l a r l y , f o r the d i p o l e resonance of a c a v i t y , we f i n d k a = (12u/(A+6p)]

(24)

L

In t h e case o f a s o l i d i n c l u s i o n , the d i p o l e resonance f r e q u e n c y i s g e n e r a l l y lower than the monopole resonance f r e q u e n c y . For low shear modulus m a t e r i a l w i t h dense p a r t i c l e s , Chaban f i n d s the d i p o l e resonance frequency 1/2

a p

2

P j

(25)

+p

R e c e n t l y we have c a l c u l a t e d t h i s term when the shear r i g i d i t y o f t h e m a t r i x m a t e r i a l i s taken i n t o account. We f i n d the d i p o l e resonance frequency 1

A + 2p

)

m

m ^ ) - ^ % 9 ) - ^ p ^

(26)

p r o v i d e d the i n c l u s i o n m a t e r i a l i s r i g i d and f a i r l y i n c o m p r e s s i b l e . Here g(P ,p)=p;;(2 + p ) + 9 P + ( l + 2 p ) 0

0

(27)

Where p ^ = p/(A+2p) and p = p l / p . E q u a t i o n 26 reduces to Equat i o n 25 i n the l i m i t pn-^O. To f i r s t o r d e r i n po, E q u a t i o n 26 becomes 0

(28) 0

a

p

2 +p P ]

0

2 j+p P

F i n a l l y , we note that K u s t e r and Toksoz (20) i n t r o d u c e d a new method i n which the displacement f i e l d s , expanded i n s e r i e s , f o r waves s c a t t e r e d by an " e f f e c t i v e " composite medium and i n d i v i d u a l

236


i n c l u s i o n s are equated. The c o e f f i c i e n t s of the s e r i e s expansions of the displacement f i e l d s p r o v i d e r e l a t i o n s h i p s between the e l a s t i c moduli of the e f f e c t i v e medium and those of the m a t r i x and the inclusions. Both s p h e r i c a l and s p h e r o i d a l i n c l u s i o n s were c o n s i d ered. They d e r i v e d t h e i r model o n l y f o r the long wavelength l i m i t and, i n that l i m i t , t h e i r r e s u l t s f o r s p h e r i c a l i n c l u s i o n s are i d e n t i c a l to the s t a t i c theory of Kerner. R e c e n t l y , attempts (21) have been made to bypass the low frequency l i m i t a t i o n of the K u s t e r Toksoz method. Although w e l l known r e s u l t s were o b t a i n e d f o r the case of a i r bubbles i n water, g e n e r a l l y the r e s u l t s are b e l i e v e d to be f o r t u i t o u s as the assumptions made i n dropping the term cont a i n i n g the r a d i u s of the e f f e c t i v e s c a t t e r e r have r e c e n t l y been q u e s t i o n e d (22,23). A l l of the above r e s u l t s f o r M and the v a r i o u s resonance f r e q u e n c i e s are c o n t a i n e d i n the long wavelength, low c o n c e n t r a t i o n e x p r e s s i o n f o r the e f f e c t i v e compressional wave v e c t o r d e r i v e d from the Foldy-Twersky theory f o r the coherent e f f e c t i v e f i e l d (24-26), namely k* = k + (3(J>/2ka )f = oo/c* = (oo/c) + ia 3

(29)

where f i s the forward s c a t t e r i n g amplitude of an i s o l a t e d scatterer. T h i s equation c o n t a i n s c o n t r i b u t i o n s from a l l resonances of the s c a t t e r e r s v i a f . I t i s perhaps most r e l i a b l e at f r e q u e n c i e s below the lowest resonance. In s p e c i a l s i t u a t i o n s i t i s s u f f i c i e n t to c o n s i d e r o n l y the lowest resonance at low f r e q u e n c i e s , e i t h e r the monopole or the d i p o l e . The r e s u l t s then are e q u i v a l e n t to the e x p r e s s i o n s f o r M quoted above. In g e n e r a l , at l e a s t s e v e r a l r e s o nances should be r e t a i n e d i n Equation 29 s i n c e t h e i r c o n t r i b u t i o n s usually overlap. As co approaches z e r o , the s t a t i c l i m i t , d i s c u s s e d i n the p r e v i o u s s e c t i o n , i s reproduced at low c o n c e n t r a t i o n s of inclusions. I f o t h e r than s p h e r i c a l s c a t t e r e r s or d i s t r i b u t i o n s of s c a t t e r e r s i z e s and o r i e n t a t i o n s are t o be c o n s i d e r e d , f / a ^ should be r e p l a c e d by i t s average v a l u e . The appearance of the forward s c a t t e r i n g amplitude i n Equat i o n 29 enables us to g i v e a c o n c e p t u a l l y a p p e a l i n g i n t e r p r e t a t i o n of the a t t e n u a t i o n c o e f f i c i e n t a s s o c i a t e d w i t h k*. Using the forward s c a t t e r i n g theorem, which i s v a l i d f o r any s c a t t e r e r geometry, one f i n d s (27-30) lmag(0/k = (opp + ops + oa)/4n

(30)

Opp i s the l o n g i t u d i n a l to l o n g i t u d i n a l or "p to p-wave" s c a t t e r i n g c r o s s s e c t i o n of an i s o l a t e d s c a t t e r e r . At low f r e q u e n c i e s i t i s p r i m a r i l y a s s o c i a t e d with the monopole s c a t t e r i n g mode. O p i s the mode c o n v e r s i o n or l o n g i t u d i n a l to shear wave or "p to s-wave" c r o s s section. I t i s p r i m a r i l y a s s o c i a t e d with the d i p o l e mode at low frequencies. o i s the a b s o r p t i o n c r o s s s e c t i o n which d e s c r i b e s energy d i s s i p a t i o n i n s i d e v i s c o e l a s t i c i n c l u s i o n s (embedded i n an o t h e r w i s e e l a s t i c m a t r i x m a t e r i a l ) . I t i s a c t i v e i n a l l modes of excitation. Combining Equation 29 and 30 y i e l d s S

a

2

3

c=c(l+34> Real ( f ) / 2 k a )

- 1

(31)


237


and a = 34>(opp + ops -I- oa)/8na

(32)

The e f f e c t i v e a t t e n u a t i o n c o e f f i c i e n t i s t h e r e f o r e p r o p o r t i o n a l t o the t o t a l ( e x t i n c t i o n ) s c a t t e r i n g c r o s s s e c t i o n , o r e q u i v a l e n t l y , t o the f r a c t i o n of the energy f l u x d e n s i t y s c a t t e r e d i n a l l d i r e c t i o n s (opp Ops) P u n i t d i s t a n c e t r a v e l l e d , plus the f r a c t i o n c o n v e r t e d i n t o heat by the i n c l u s i o n s ( o ) . Other c l o s e d form e x p r e s s i o n s f o r k* e x i s t (31.32). These reduce the E q u a t i o n 29 at low f r e q u e n c i e s and i n the l i m i t o f low c o n c e n t r a t i o n s of i n c l u s i o n s . The forward s c a t t e r i n g amplitude p l a y s an important r o l e i n such t h e o r i e s (see a l s o Twersky ( 2 5 ) . where m u l t i p l e s c a t t e r i n g i s taken i n t o a c c o u n t ) . More fundam e n t a l l y , i t i s the unperturbed s i n g l e p a r t i c l e resonance spectrum which i s m i r r o r e d i n k* v i a f . In the case of h i g h concentrations of i n c l u s i o n s , c o r r e l a t e d motions of s c a t t e r e r s may become important and dynamic l o a d i n g may a l t e r the s p e c t r a to some e x t e n t . Such e f f e c t s a r e p i c k e d up by t h e o r i e s which e x p l i c i t l y attempt t o t r e a t near f i e l d s and m u l t i p l e s c a t t e r i n g i n t e r a c t i o n s [ 5 ] . The r e s u l t i n g e x p r e s s i o n s f o r k* a r e i m p l i c i t and depend on a p p r o p r i a t e l y m o d i f i e d s i n g l e p a r t i c l e s p e c t r a , r a t h e r than the ( f a r f i e l d ) a p p r o p r i a t e l y m o d i f i e d s i n g l e p a r t i c l e s p e c t r a , r a t h e r than the forward s c a t t e r i n g amplitudes of i s o l a t e d s c a t t e r e r s . At low c o n c e n t r a t i o n s of i n c l u s i o n s , E q u a t i o n 29 should be y i e l d a c c u r a t e p r e d i c t i o n s . Expanding Z t o f i r s t order i n [ 3 and f a r e s m a l l , s i n c e f does not depend on (J). +

e r

a

There a r e i n d i c a t i o n s that c o r r e l a t i o n i n s c a t t e r e r responses and i n t e r a c t i o n s between s c a t t e r e r s may not be important g e n e r a l l y , even a t moderately high volume c o n c e n t r a t i o n s of s c a t t e r e r s (33.34). E q u a t i o n s l i k e Equation 29 should t h e r e f o r e y i e l d u s e f u l q u a l i t a t i v e i n f o r m a t i o n about the low frequency s t r u c t u r e i n Z and a over extended ranges of values of 4>, depending i n the p a r t i c u l a r system under c o n s i d e r a t i o n . A study of the c r o s s s e c t i o n s adds f u r t h e r i n f o r m a t i o n about the types of waves s c a t t e r e d ( l o n g i t u d i n a l o r s h e a r ) and the r e l a t i v e importance of energy d i s s i p a t i o n by the inclusions. One f i n d s that s o l i d i n c l u s i o n s tend to d e f i n e the low frequency s t r u c t u r e i n k*, the a t t e n u a t i o n edge and the resonance i n c, by means o f the d i p o l e resonance, whereas c a v i t i e s tend to d e f i n e i t by means of the monopole resonance ( 3 5 ) . Since the d i p o l e resonance of a s o l i d i n c l u s i o n i n v o l v e s c e n t e r o f mass motion of the i n c l u s i o n , t h i s resonance tends to be at lower f r e q u e n c i e s than t h a t due to the monopole of a c a v i t y which merely i n v o l v e s c a v i t y w a l l motion. The h e a v i e r the i n c l u s i o n s i n the case o f s o l i d i n c l u s i o n and the s o f t e r the matrix m a t e r i a l i n the case of c a v i t i e s , the lower i n frequency i n which s i g n i f i c a n t a t t e n u a t i o n i s a c h i e v e d ( a t t e n u a t i o n edge) and the l a r g e r the c o r r e s p o n d i n g s c a t t e r i n g c r o s s sections. In r i g i d m a t r i x m a t e r i a l s , which support shear waves because o f t h e i r shear r i g i d i t y , mode c o n v e r s i o n s c a t t e r i n g by heavy s o l i d i n c l u s i o n s predominates a t low f r e q u e n c i e s ( o p » O p p ) . In t h a t case a reduces to S

a — 34>(ops + oa)/8na

(33)

238


F i g u r e 1 shows an example; 30 percent l e a d i n c l u s i o n s (a = 2mm) i n epoxy, Epon 828Z. Here o — 0 and a i s seen to be e s s e n t i a l l y due to o a t low f r e q u e n c i e s . The volume c o n c e n t r a t i o n of i n c l u s i o n s i n t h i s example i s , however, known to be r a t h e r h i g h f o r E q u a t i o n (29) t o y i e l d exact p r e d i c t i o n s . I t has been p o i n t e d out t h a t E q u a t i o n (31) does not p r e d i c t the s h i f t i n resonance f r e q u e n c y w i t h c o n c e n t r a t i o n that i s e x p e r i m e n t a l l y observed f o r Pb i n epoxy at volume c o n c e n t r a t i o n s above 5 p e r c e n t . Note the steep r i s e of of and the a t t e n u a t i o n edge ( p r e d i c t e d by E q u a t i o n (30) a t a l l c o n c e n t r a t i o n s of i n c l u s i o n s ) , approaching the c h a r a c t e r i s t i c 0)4 ( R a y l e i g h ) dependence of o below the d i p o l e resonance f r e q u e n c y ; E q u a t i o n ( 2 6 ) . At ka>l, a t t e n u a t i o n i s c o n t r o l l e d by h i g h f r e q u e n c y resonances above the quadrupole resonance. a

ps

ps

S o f t m a t r i x m a t e r i a l s do not support shear waves and c o n s e q u e n t l y d i p o l e s c a t t e r i n g by s o l i d i n c l u s i o n s and mode c o n v e r s i o n e f f e c t s are weak i n such m a t e r i a l s . The c o r r e s p o n d i n g resonance frequency i s g i v e n by Equation ( 1 9 ) . Mode c o n v e r s i o n i s a l s o weak i n the case of c a v i t i e s i n s o f t m a t e r i a l s . The d i p o l e resonance frequency i s g i v e n by Equation ( 2 8 ) . In the case of c a v i t i e s i n s o f t m a t e r i a l s , monopole s c a t t e r i n g predominates a t low frequencies. The lowest resonance frequency i s g i v e n by E q u a t i o n (18) and a reduces to

a^34>(o

PP

+ oa)/8na

3

(34)

An example i s shown i n F i g u r e 2 f o r 30 percent v o i d s i n an e l a s t i c rubber m a t e r i a l ( p c = 3.04 x 10^ N/m , p" = p = 1.13 x 10° N/m , p = 1.19 x 1 0 Kg/m , a = 2mm) Note that mode c o n v e r s i o n i s v i r t u a l l y absent and the a t t e n u a t i o n edge v a r i e s as o>^, as i n F i g u r e 1. In a v i s c o e l a s t i c m a t e r i a l one would observe a t r a n s i t i o n from the co^ dependence to a more gradual v a r i a t i o n a t low f r e q u e n c i e s (a/k a )fl

1/2

(41)

E q u a t i o n 29 and o t h e r c l o s e d form s o l u t i o n s f o r k* (31,32) g i v e v e r y s i m i l a r r e s u l t s ; note the s t r u c t u r e i n a(30 percent Pb) i n F i g u r e 1. The d i p o l e resonance frequency i s g i v e n by Equation 20. F i n a l l y , we a l s o note t h a t a t lower c o n c e n t r a t i o n s of Pb, agreement between experiment and t h e o r y i s much improved ( 3 9 ) . Summary We have reviewed v a r i o u s r e s u l t s i n the theory of the e f f e c t i v e m a t e r i a l parameters of inhomogeneous m a t e r i a l s with random d i s t r i b u t i o n s of s p h e r i c a l i n c l u s i o n s . L i m i t a t i o n s to low volume c o n c e n t r a t i o n s have been s t r e s s e d r e p e a t e d l y although ranges of a p p l i c a b i l i t y a r e expected to vary from system t o system. In the case of m i c r o s c o p i c v o i d s i n rubbery m a t e r i a l s , the range of a p p l i c a b i l i t y of K e r n e r s q u a s i - s t a t i c parameters appears t o be l a r g e r than commonly assumed. As the r a d i i of the i n c l u s i o n s a r e i n c r e a s e d and resonances encountered, the s t r u c t u r e o f the e f f e c t i v e parameters becomes more p r o b l e m a t i c . A q u a l i t a t i v e a n a l y s i s , based on the s i n g l e p a r t i c l e forward s c a t t e r i n g amplitude which o c c u r s i n c l o s e d form s o l u t i o n s f o r the e f f e c t i v e wave v e c t o r , i s p o s s i b l e , however. S t u d i e s of the lowest resonances and t h e i r dependence on m a t e r i a l combinations y i e l d u s e f u l i n f o r m a t i o n about the a t t e n u a t i o n edge, wave speed, and s c a t t e r i n g c r o s s s e c t i o n s as w e l l as s c a t t e r i n g and a b s o r p t i o n mechanisms. At l a r g e volume c o n c e n t r a t i o n s n u m e r i c a l approaches must be used. The t r a n s i t i o n from e l a s t i c t o v i s c o e l a s t i c m a t e r i a l s i s achieved by simply i n t r o d u c i n g complex v a l u e d , c a s u a l l y r e l a t e d m a t e r i a l parameters. Arbitrary inclusion shapes, s i z e s , and o r i e n t a t i o n s may i n p r i n c i p a l be t r e a t e d once the s i n g l e p a r t i c l e resonance s p e c t r a a r e known. 1

244


F i g u r e 5. C o m p a r i s o n o f M e a s u r e d a n d C a l c u l a t e d Wave S p e e d s ; 1 5 % Pb i nE p o x y . ( R e p r o d u c e d w i t h p e r m i s s i o n f r o m r e f . 3 9 . C o p y r i g h t 1983 A m e r i c a n I n s t i t u t e o f P h y s i c s . )

12.

MADIGOSKY AND SCHARNHORST


245

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6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Ferry, J., Viscoelastic Properties of Polymers, John Wiley and Sons, New York, 1970, p. 314. Madigosky, W. and Lee, G., "Automated Dynamic Young's Modulus and Loss Factor Measurements," J. Acoust. Soc. Am., 1979, 66, pp. 345-349. Madigosky, W. and Lee, G., "Improved Resonance Technique for Materials Characterization," J. Acoust. Soc. Am., 1983, 73, pp. 1374-1379. Madigosky, W., "The Effects of Crosslinking on the Dynamic Mechanical Properties of Urethanes," Rubber World, 1987, 196, No. 6, pp. 27-29. Varadan, V. K., "Multiple Scattering of Acoustic, Electromagnetic and Elastic Waves, Acoustic, Electromagnetic and Elastic Wave Scattering-Focus on the T-Matrix Approach, Pergamon Press, New York, 1980, pp. 103-134. Urick, R. J., "A Sound Velocity Method for Determining the Compressibility of Finely Divided Substances," J. Appl. Phys., 1947, 18, pp. 983-987. Madigosky, W. M. and Bradley, D. L., "Construction and Focusing Properties of a Solid Spherical Acoustic Lense," NOL TR 68-145, Silver Spring, MD, Nov 1968. Kerner, E. H., "The Elastic and Thermo-elastic Properties of Composite Media, Proc. Phys. Soc. London, B, 1956, 69, pp. 808813. DeRuntz, T. A., Jr., "Micromechanics and Macromechanics of Syntactic Foams," LSMC Report 6-78-68-44, Lockheed Palo Alto Research Laboratories, Sep 1968. van der Pol, C., "On the Rheology of Concentrated Dispersions, Rehol. Acta, 1958, 1, pp. 198-203. Dewey, J. M., "The Elastic Constants of Materials Loaded with Non-Rigid Fillers," J. Appl. Phys., 1946, 18, pp. 578-581. Christensen, R. M., Mechanics of Composite Materials, John Wiley and Sons, New York, 1979, pp. 31-72. Hashin, Z., "The Elastic Moduli of Heterogeneous Materials," J. Appl. Mech., 1962, 29, pp. 143-149. Honig, E. P.; Wierenga, P. E., and van der Linden, J. H. M., Theory of elastic behavior of composite materials, J. Appl. Phys., 1987, 62, pp. 1610-1612. van der Linden, J. H. M.; Wierenga, P. E.; Honig, E. P., "Viscoelastic Behavior of Polymer Layers with Inclusions," J. Appl. Phys., 1987, 62, pp. 1613-1615. Chaban, I. A., "Calculation of the Effective Parameters of Micro-Inhomogeneous media by the Self-Consistent Field Method," Sov. Phys.-Acoust., 1965, 11, PP. 81-86. Meyer, E.; Brendel, K.; and Tamm, K., "Pulsation Oscillations of Cavities in Rubber," J. Acoust. Soc. Am., 1958, 30, pp. 1116-1124. Gaunaurd, G.; Scharnhorst, K. P., and Uberall, H., "Giant Monopole Resonance in the Scattering of Waves from Gas-Filled Spherical Cavities and Bubbles," J. Acoust. Soc. Am., 1979, 65, pp. 573-594.

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SOUND AND VIBRATION DAMPING WITH POLYMERS Kligman, R.; Madigosky, W.; and Barlow, J., "The Effective Dynamic Properties of Composite Viscoelastic Materials," J. Acoust. Soc. Am., 1981, 70, pp. 1437-1444. Kuster, G. T. and Toksoz, M. N., "Velocity and Attenuation of Seismic Waves in Two-Phase Media: Part 1. Theoretical Formulations; and Part 2. Experimental Results," Geophysics, 1974, 39, pp. 587-618. Gaunaurd, G. C. and Uberall, H., "Resonance Effects and Ultrasonic Effective Properties of Particulate Composites," J. Acoust. Soc. Am., 1983, 74, pp. 305-313. Scharnhorst, K. P., "Comments on the Applicability of the Kuster-Toksoz Method to the Derivation of the Dynamic Material Parameters of Inhomogeneous Media," J. Acoust. Soc. Am., 1987, 82, pp. 692-695. Anson, L. W. and Chivers, R. C., "Ultrasonic Propagation in Suspensions -- A Comparison of a Multiple Scattering and Effective Media Approach," J. Acoust. Soc. Am., 1989, 85, pp. 535-554. Foldy, L. O., "The Multiple Scattering of Waves," Phys. Rev., 1945, 67, pp. 107-119. Twersky, V., "On Propagation in Random Media of Discrete Scatters," Proc. Am. Math. Soc. Sym. Stochas. Proc. Math. Phys. Eng., 1964, 16, pp. 84-116. Ishimaru, A., Wave Propagation and Scattering in Random Media, Academic Press, New York, 1978, Vol. 2, Chap. 14. Varatharajulu, V. and Pao, Yih-Hsuig, "Scattering Matrix for Elastic Waves, I. Theory," J. Acoust. Soc. Am., 1976, 60, pp. 556-566. Bostroem, A., "Multiple Scattering of Elastic Waves by Bounded Obstacles," J. Acoust. Soc. Am., 1980, 67, (2), pp. 399-413. Lim, R. and Hackman, R. H., "Fundamental Analysis of a Novel Composite Sound Attenuator Using the T-Matrix Approach," NCSC Technical Report U2120-88-40, 1988. Scharnhorst, K. P.; Hackman, R. H.; and Lim, R., "Damping Via Mode Conversion," Proceedings of the ACS Division of Polymeric Materials: Science and Engineering, 1989, 60, pp. 497-501. Lax, M., "Multiple Scattering of Waves," Rev. Mod. Phys., 1951, 23, pp. 287-310. Waterman, P. C. and Truell, R., "Multiple Scattering of Waves," J. Math. Phys., 1961, 2, pp. 512-537. Varadan, V. V. and Varadan, V. K., "Configuration with Finite Number of Scatterers -- A Self Consistent T-Matrix Approach," J. Acoust. Soc. Am., 1981, 70, pp. 213-217. Domany, E.; Eutin-Wohlman, O.; and Mizrachi, L., "Multiple Scattering Formalism: Application to Scattering by Two Spheres," J. Appl. Phys., 1984, 56, pp. 132-136. Scharnhorst, K. P.; Madigosky, W. M.; and Balizer, E., "Scattering Coefficients and the Absorption Edge of Longitudinal Coherent Sound Waves in Selected Inhomogeneous Materials," NSWC Technical Report 85-196, Silver Spring, MD, Jun 1985. Kinra, V. K., "Dispersive Wave Propagation in Random Particulate Composites, Recent Advances in Composites in the U.S. and Japan," ASTM STP 864, 1985, pp. 309-325.

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37. Madigosky, W. and Warfield, R., "Magnitude of the Ultrasonic Volume Viscosity," Acustica, 1984, 3, pp. 3470-3474. 38. Rogers, L., "Operators and Fractional Derivatives for Viscoelastic Constitutive Equations," J. Rheol., 1983, 27, pp. 351372. 39. Sayers, C. M. and Smith R. L., "Ultrasonic Velocity and Attenuation in an Epoxy Matrix Containing Lead Inclusions," J. Phys. D: Appl. Phys., 1983, 16, pp. 1189-1194. RECEIVED January 24, 1990

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