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ULTRASONIC CHARACTERIZATION OF
POROSITY USING THE KRAMERS-KRONIG
RELATIONS
Jérôme Rose, D. Hsu, L. Adler
To cite this version:
ULTRASONIC CHARACTERIZATION OF POROSITY USING THE KRAMERS-KRONIG RELATIONS
J.H. ROSE, D.K. HSU AND L. ADLER+
Ames Laboratory, USDOE, Iowa State University, Ames, Iowa 50011, U.S.A.
+Welding Engineering Department, Ohio State University, Columbus, Ohio 43210, U. S.A.
Abstract
-
A new a l g o r i t h m i s proposed t o determine t h e volume f r a c t i o n o f pores i n s o l i d s using the frequency dependent u l t r a s o n i c attenuation. The a l g o r i t h m was developed by examining the Kramers-Kronig r e l a t i o n between the p o r o s i t y induced u l t r a s o n i c a t t e n u a t i o n and the change i n sound v e l o c i t y . The method i s t e s t e d using data measured f o r several porous aluminum samples.I
-
INTRODUCTIONThis paper i s s t r u c t u r e d as f o l l o w s . F i r s t , a model i s given which r e l a t e s t h e p o r o s i t y induced a t t e n u a t i o n and the s h i f t i n t h e sound speed. Algorithms f o r the volume f r a c t i o n , c, and the average pore size, ti, are then derived. F i n a l l y , these algorithms are t e s t e d and found t o be s a t i s f a c t o r y using experimental attenu- a t i o n s measured i n porous c a s t aluminum samples.
P o r o s i t y i s modeled as a uniform random d i s t r i b u t i o n o f spherical voids o f various r a d i i i n an otherwise homogeneous and i s o t r o p i c e l a s t i c s o l i d . The volume frac- t i o n i s assumed small (c
-.
0,
p r a c t i c a l l y c<
5%). Following recent treatments o f t h i s model ( i n p a r t i c u l a r Ref.I ) ,
we o b t a i nand
Here a, A V and V o are r e s p e c t i v e l y the attenuation, the v e l o c i t y s h i f t and, the v e l o c i t y o f l o n g i t u d i n a l sound i n pore f r e e m a t e r i a l , f o r a l o n g i t u d i n a l l y p o l a r i z e d displacement f i e l d . Also n(a)da denotes the number o f pores p e r u n i t volume w i t h r a d i i between a and' atda. The wave vector i s denoted by
k.
A(k,a) denotes the l o n g i t u d i n a l t o l o n g i t u d i n a l forward s c a t t e r i n g amplitude.C10-788 JOURNAL DE PHYSIQUE
The expression o f c a u s a l i t y i n t h e frequency domain (Kramers-Kronig re1ations)Z imp1 i e s
and a s i m i l a r r e l a t i o n f o r ~ m ~ ( k ) / k ~ . The s l a s h i n d i c a t e s the p r i n c i p a l p a r t o f t h e i n t e g r a l . S u b s t i t u t i o n o f ( 3 ) i n ( 2 ) y i e l d s
an expression f a m i l i a r from o t h e r contexts; see f o r example Ref. 3.
The l o n g wavelength l i m i t o f t h e v e l o c i t y s h i f t [Eq. ( 4 ) l leads t o o u r basic r e s u l t . As k + 0, Eq. (4) becomes
On i n t u i t i v e grounds AV(k + 0 ) i s ( 1 ) expected t o depend 1 in e a r l y on c as c + 0 and ( 2 ) t o be independent o f t h e pore s i z e d i s t r i b u t i o n . Consequently Eq.
( 5 ) can be converted i n t o an a l g o r i t h m f o r determining c from a ( k ) .
We proceed by e v a l u a t i n g Eq. ( 2 ) d i r e c t l y i n t h e long wavelength l i m i t . As k + 0, ReA(k,a) = a ( ~ p k ~ a ~ + o(k4a4)). Here A2 i s a d i m e n s i ~ n l e s s expansion c o e f f i c i e n t which depends o n l y on t h e Poisson r a t i o o f the h o s t m a t e r i a l . An a n a l y t i c expression f o r Ap can be obtained from Ref. 4. S u b s t i t u t i o n o f t h e k -+ 0 form o f ReA(k) i n Eq. ( 2 ) y i e l d s
This i s o u r basic algorithm. I t determines c from a ( k ) i n a way which does n o t depend e x p l i c i t l y on t h e pore s i z e d i s t r i b u t i o n .
Once c has been determined from Eq.
(6),
a rough estimate o f t h e average pore s i z e can be obtained. A t h i g h frequencies ( k a > > l ) the a t t e n u t a t i o n can be computed from an a c o u s t i c r a y p i c t u r e . I t becomes a ( k + m) = N a <a2>. Here N i s the t o t a l number o f pores p e r u n i t volume. The average cross-sectional area p e r pore i s given by r<a2>; where<.
.
.>
denotes t h e expectation value over t h e s i z e d i s t r i b u - t i o n . On t h e o t h e r hand c = 471 ~ < a ~ > / 3 . Combining those r e s u l t s y i e l d s the estimate I = <a3>/<a2> = 3C/(4 a(k -+ m)). We expect ti t o p r o v i d e a reasonable estimate f o r t h e radius, i f t h e d i s t r i b u t i o n i s s h a r p l y peaked about a mean value. EXPERIMENTAL RESULTSFrequency ( MH3)
Fig. 1A. Shows experimental ( - - - ) and t h e o r e t i c a l a t t e n u a t i o n
(--) o f Sample #1510.
Frequency, f (MH31
F i g . 1B. Shows i n t e g r a n d i n Eq. ( 6 ) . V e r t i c a l dashed l i n e s i n d i c a t e minimum and maximum usable frequencies (#1510).
Table 1. Gives sample #, a c t u a l c ( d e n s i t y measurement),
c
( u l t r a s o n i c experiment), and r a d i u s estimate,Z.
Figure (1A) shows t h e experimental a t t e n u a t i o n measured f o r sample 1510. F i g u r e (1B) shows a p l o t of a(k)/k2; the i n t e g r a n d i n Eq. ( 6 ) . The e v a l u a t i o n o f t h e i n t e g r a l from 0 t o - r e q u i r e s t h a t t h e data be extended using p r i o r i n f o r m a t i o n . The known asymptotic form o f a ( k ) leads us (1) t o s e t a ( k ) = a(kmax) f o r wavevectors k > (kmax); and ( 2 ) t o l e t a ( k ) s k4 f o r k l e s s than the s m a l l e s t usable wavevector. These extensions t y p i c a l l y account f o r 20% o f t h e t o t a l i n t e g r a l . F i n a l l y A2 = .57 f o r A357 aluminum a l l o y .
C10-790 JOURNAL DE PHYSIQUE
ACKNOWLEDGEMENT
This work was sponsored by the Center f o r Advanced Nondestructive Evaluation, operated by t h e Ames Laboratory, USDOE, f o r t h e A i r Force Wright Aeronautical Laboratories/ M a t e r i a l s Laboratory under Contract No. W-7405-ENG-82 w i t h Iowa State U n i v e r s i t y . REFERENCES
/l/Gubernatis, J. E. and Domany, E., Review o f Progress i n Q u a n t i t a t i v e Nondestruc- t i v e Evaluation,
A,
833 (1983)./2/Newton, R. G., S c a t t e r i n g Theory o f Waves and P a r t i c l e s , (Springer-Verlag, 1982). /3/Beltzer, A. I. and Brauner, N., J. Acoust. Soc. Am. 76, 962 (1984).
/4/Ying, C. F. and T r u e l l , R., 3. Appl. Phys.
