[IEEE IEEE 1985 Ultrasonics Symposium - San Francisco, CA, USA (1985.10.16-1985.10.18)] IEEE 1985...

PMSE SPECTROSCOPY I N LOSSY MEDIA

J. F. Muratore and H. R. Carleton

College o f Engineering and Applied Sciences State University o f New York, Stony Brook, N.Y. 11794

ABSTRACT

We have approached t h e problem of accw r a t e l y measuring the d i s p e r s i o n inherent i n t h e phase angle , phase v e l o c i t y , and group v e l o c i t y i n l o s s y materials using d i g i t a l processing tech- niques on broadband u l t r a s o n i c s i g n a l s i n the range 0 - 20 MHZ through porous g r a p h i t e and metallic alloys wi th micros t ruc ture . using FFT a l g o r i t h m and H i l b e r t t ransforms to g e n e r a t e both total phase an3 minimm phase spectra, we calcw late t h e group and phase v e l o c i t y spectra with high p r e c i s i o n and over a wide frequency band. Useful information on d i s p e r s i o n i n both t h e lossy porous medium and the denser a l l o y is determined. The na ture of t h e d i s p e r s i o n is shown to be a n important tal i n determining the mechanisms causing a t t e n u a t i o n and t h e r e l a t i o n s h i p of the s c a t t e r i n g to a t t e n u a t i o n .

I n t r d u c t i o n

The technique of u l t r a s o n i c spectroscopy is a va luable tool i n t h e c h a r a c t e r i z a t i o n of material micros t ruc ture and t h e i n t e r n a l phys ica l processes t h a t lead to energy lossl-2. Much work has been done with t h e spectra of dense, nonporous materials such as p l y c r y s t a l l i n e metals, a l l o y s , and amorphous p l a s t i c s . Due to l o w energy loss, good s i g n a l - t e n o i s e ratios are achieved and use- f u l information over a wide frequency band can be obta ined , Our i n v e s t i g a t i o n s have been concerned wi th lossy m a t e r i a l s , such as porous and ceramic m e d i a , t h a t e x h i b i t a high a t t e n u a t i o n , where we have found t h a t a c c u r a t e spectra can be generated Over a wide bandwidth with p r e c i s i o n d i g i t a l sampling and s i g n a l processing technique^^-^.

We have obtained spectra of f l u i d - f i l l e d porous g r a p h i t e t h a t e x h i b i t as much as 550 dB/m a t t e n u a t i o n Over a 12 MHz range i n frequency f o r a total dynanic range of 63 dB. I n this report, we show t h a t it is u s e f u l to s tudy t h e phase spectra i n conjunct ion wi th the anpl i tude spectra and t h a t t h e phase spectra may r e v e a l a d d i t i o n a l informa- t i o n about t h e inherent processes o f a t tenua- t ion . Ran t h e w r k of Bode and other^^-^, it is clear t h a t the amplitude and phase of a s y s t e n

0090-5607/85/0000-1047 $1.00 6 1985 IEEE

t r a n s f e r func t ion are i n t i m a t e l y related. We w i l l demonstrate, however, t h a t a d d i t i o n a l cwntribu- t i o n s to t h e d i s p e r s i o n o f a s y s t a n can appear i n processes which are not f u l l y accessible f r a n Kramers-Kronig ana lys i s . Frequency-dependent a t t e n u a t i o n is n e c e s s a r i l y acccmpanied by v e l o c i t y d i s p e r s i o n which can best be der ived frcm t h e c a n p l e t e phase spectrum and represented by a r e f r a c t i v e index r e l a t e d to the sound v e l o c i t y i n t h e pure material.

Using s i g n a l ana lys i s9 , it can be shown that a system t r a n s f e r func t ion H ( U ) can be represented as the product of t h r e e parts,

H ( U ) = & ( U ) ' h 1 ( W ) . H a ( U ) (1)

Since each c a n p n e n t of t h i s expression can be w r i t t e n i n canplex exponent ia l form, t h e total phase of t h e s y s t m is the sum of three canponent phases and can be w r i t t e n as

( 2 )

The mininun-phase term, 4 m, r e p r e s e n t s t h e phase t h a t is r e l a t e d to t h e a t t e n u a t i o n through the Kramers-Kronig r e l a t i o n s . T h i s term has k e n widely appl ied to t h e a n a l y s i s of t h e a t t e n u a t i o n of l i n e a r , c a u s a l sys tans . The a l l -pass phase c o n t r i b u t i o n , @ a cannot be derived f r a n a H i l b e r t t ransform of the a t t e n u a t i o n spectm and hence provides a d d i t i o n a l information on the processes tak ing p lace i n t h e material. W show that such a phase c a n p n e n t does e x i s t i n porous g r a p h i t e and i n an a l l o y wi th a n inhancgenous micros t ruc ture . mr r e s u l t s show that s c a t t e r i n g mechanisms are charac te r ized by this s i g n i f i c a n t non-minimm- phase c o n t r i b u t i o n and must be s tudied f r a n the phase spectra. It also fol lows that ana lyses of a t t e n u a t i o n processes t h a t assune o n l y a minimum- phase spec t run are s u b j e c t to s e r i o u s error when s c a t t e r i n g is present .

The @d term is a pure-delay m p o n e n t a r i s i n g f r a n t h e time d e l a y of t h e pulse t r a v e l i n g through t h e medim and can only be der ived f r a n t h e phase spectnm when d i s p e r s i o n is present . S ince the shape o f t h e pulse changes as it t r a v e r s e s t h e medium, t i m e d e l a y measurenents i n the time danain are ambiguous.

1985 ULTRASONICS SYMPOSIUM - 1047

Our emphasis i n t h i s report is to show t h a t the phase spec t run as c a l c u l a t e d wi th t h e amplitude spec t run through FFT and deconvolut ion a lgor i thms can be separated i n t o its m p o n e n t parts and t h a t such decanposi t ion adds f u r t h e r i n s i g h t i n t o t h e na ture of t h e i n t e r n a l phys ica l processes t h a t mani fes t themselves i n the response of t h e system to propagating sound waves.

Method

The l o s s y m a t e r i a l used was an extruded porous g r a p h i t e UF-4S (Ultra Carbon Corp., Bay C i t y , M I ) wi th a maximm g r a i n s i z e of 203 u m , as reported by the manufacturer. A d i s k was c u t fKan a 5.08 cm d i a n e t e r rod and ground and lapped down to a th ickness of 0.115 cm. A f t e r u l t r a s o n i c c leaning to clear t h e pres , t h e sample was s a t u r a t e d i n d i s t i l l e d water using a vacuun impregnation technique to achieve as close to f u l l s a t u r a t i o n as p s s i b l e . The average s i z e of t h e pres is 100 u m and they account f o r about 20% of t h e volune of t h e mater ia l .

I n o r d e r to canpare t h e r e s u l t s i n a porous m a t e r i a l wi th a material which e x h i b i t s a d i f f e r e n t s c a t t e r i n g mechanisn, t h e sane experimental procedures and a n a l y s i s were p r f o r m e d on a metallic a l l o y . We chose ZA-12, an as-cast a l l o y , Zn(88)Al(12) , l i s t e d by I L Z d 0 , wi th d e n s i t y 5.913 g/cm3. Measurements were made on a rec tangular block of 2.54 cm thickness .

The u l t r a s o n i c spectroscopy system is depic ted i n Fig.1. We use a through-transniss ion imnersion technique with tw Panametrics (Waltham, MA) model V-312 10 MHz wideband t ransducers mounted i n a rotatable assenbly. With t h e t r a n s n i t t e r and r e c e i v e r a l igned a1or-q a s t r a i g h t t ransmission pa th , t h e s m p l e g r a p h i t e d i s k , he ld i n an absorbing a p e r t u r e to he lp e l i m i n a t e k a m spreading problems and r a d i a t i v e coupl ing, is o r i e n t e d with its f a c e s normal to the u l t r a s o n i c beam by maximizing a pulse echo s i g n a l r e f l e c t e d o f f t h e f r o n t f a c e of t h e sample. To g e t t h e broades t p o s s i b l e useable bandwidth, t h e t r a n s m i t t e r is e x c i t e d by a 50 v pulse produced by an HP 214A pulse genera tor . The s i g n a l f r a n the rece iv ing t ransducer is acquired by equiva len t t ime d i g i t a l sampling i n a Tektronix 7854 d i g i t a l storage oscilloscope, averaged 100 times to increase S/N ratio, and t r a n s f e r r e d to f i l e s t o r a g e i n a PDP 11/34 canputer , where f u r t h e r processing is performed. The frequency domain spectra are c a l c u l a t e d using a radix-2 FFT algori thm f o r both t h e s i g n a l t ransmi t ted through water a lone as a re ference and the s i g n a l t ransmi t ted with t h e sample i n s e r t e d i n t h e pa th of t h e beam. Making use of t h e deconvolution p r i n c i p l e , t h e frequency r e s p n s e ( t r a n s f e r func t ion) of t h e sample a lone is then c a l c u l a t e d by d iv id ing t h e "sample" FFT d a t a by t h e re ference (water only) spectrun. F K ~ this t r a n s f e r func t ion H(W) ,we can e x t r a c t both the anpl i tude and phase as a func t ion of frequency.

Resul t s

Fig. 2 shows t h e a t t e n u a t i o n c1 and loga spec t run obtained f o r t h e g r a p h i t e sanple . The l q ~ p l o t is convenient f o r i d e n t i f y i n g any p w e r - l a w dependence of a t t e n u a t i o n on frequency s i n c e t h e slope w i l l be propor t iona l to t h e exponent. I n t h i s case, a cons tan t exponent of 2.2 is observed over a decade range from 1 to 10 mHz. Since t h e power l a w is not an i n t e g e r , t h i s depen- dence is sugges t ive of a dis t r ibuted-parameter mechanisn i n a t t e n u a t i o n , b u t i n d i c a t e s t h e daninance of a phase scatteriy loss mechanisn t h a t e x h i b i t s f-squared behavior . This is to be expected i n a material where t h e c h a r a c t e r i s t i c dimension of t h e micros t ruc ture is of t h e same o r d e r of magnitude as t h e wavelength of t h e p r o p - g a t i n g ul t rasound. Since t h i s is t h e case with t h e porous g r a p h i t e , such behavior is reasonable.

The phase spec t run is c a l c u l a t e d by unfold ing t h e r e l a t i o n

+= a r c t a n {Im(H)/Re(H)} ( 3 )

The ambiguity i n phase which is present i n t h e a r c t a n c a l c u l a t i o n can be unmbiguously resolved by applying der ived p r o p e r t i e s of t h e phase of t h i s system a t zero frequency. S p e c i f i c a l l y , it can be shown t h a t each phase c o n t r i b u t i o n of Eq.3 must be zero f o r a through-transniss ion measurement. The minimun-phase c o n t r i b u t i o n to t h e total phase is then c a l c u l a t e d €ran t h e modulus of H(U) by using a Kramers-Kronig a l g o r i t t d l . T h i s has t h e e f f e c t of removing both pure d e l a y and a l l - p a s s phase c o n t r i b u t i o n s f r a n t h e c a l c u l a t i o n .

The e f f e c t of d i s p e r s i o n can be better understood i f t h e phase d a t a is converted t o u n i t s of r e f r a c t i v e index. T h i s is accanpl ished by f i r s t d i v i d i n g a l l phase canponents by wL, where L is sample th ickness , to o b t a i n t h e inverse phase v e l o c i t y spectnm. The zerc-frequency va lue o f the minimun-phase c o n t r i b u t i o n to t h e inverse v e l o c i t y is then subt rac ted f r a n t h a t of t h e total inverse v e l o c i t y . I t is assuned t h a t t h i s va lue of v e l o c i t y is assoc ia ted with t h e pure-delay va lue arid r e p r e s e n t s a re ference ve loc i ty . All values of t h e inverse v e l o c i t y spec t run are then d iv ided by t h e re ference inverse v e l o c i t y to o b t a i n the r e f r a c t i v e index spectrun.

The spec t run of t h e to ta l r e f r a c t i v e index is p l o t t e d as curve 4 of Fig. 3. The r e f r a c t i v e index has a va lue of 1.25 a t zero frequency and i n c r e a s e s s l i g h t l y up to 4 mHz, then decreases to 1.11 a t 20 mHz. Since a l l of t h e c o n t r i b u t i o n s to t h e r e f r a c t i v e index are a d d i t i v e , t h e pure-delay ccmponent of t h e r e f r a c t i v e index (1.0) can be s u b t r a c t e d from t h e total to o b t a i n t h e d i s p e r s i v e c o n t r i b u t i o n as p l o t t e d as graph 3 of t h i s f i g u r e and t h e minimmi-phase c o n t r i b u t i o n p l o t t e d as graph 2. It w i l l be noted t h a t t h e d i f f e r e n c e between t h e total d i s p e r s i o n and t h e minimm-phase ccmponent is the all-pass c m t r i b u t i o n to the. r e f r a c t i v e index which is p l o t t e d as graph 1. This canponent increases i n va lue with frequency anc hemes g r e a t e r than t h e minim-phase cmponent a t 1 2 mHz.

1048 - 1985 ULTRASONICS SYMPOSIUM

we have calculated the phase and group velocities fran the relations

given by Sachse and Pa012 directly fran the total phase. The phase and group velocity spectra for the graphite material are so calculated and shown in Fig. 4. Significant velocity dispersion is evident, with a larger variation being evident in the group velocity. These results indicate that substantial errors in velocity estimates of the order of 10% will be experienced if time-of-flight measurements are used to determine delay.

Similar analyses of pulse spectra fran ZA- 12 material provide an illuninating picture. The amplitude spectrun of this alloy appears in Fig. 5. "ran this, it can imnediately be seen that the range of variation of the attenuation is about 2% of that in graphite in the same frequency range, which is not surprising for a metallic material. Analysis of the log a plot shows a slope of 3.5 which indicates a mixed power law as in the graphite but closer to the fourth power dependence of frequency associated with classical Rayleigh scattering. Of greater interest is the refractive index spectra calculated in the same fashion as for the graphite and shown in Fig. 6 . Curve ( 3 ) is the phase spectrun obtained after shifting to eliminate the delay canponent. 'Ihe value of this Contribution adds only 0.5% to the total refrac- tive index. This is consistent with the decreased attenuation range.

In addition, by canparing the net phase (3) and the minim-phase ( 2 ) contributionsfit can be seen that they more closely resgnble each other and have approximately the sane range of dispersion over the frequency range resulting fran a relatively small all-pass contribution (1) below 10 mHz. Above this frequency, the all-pass contribution dominates the dispersive component of the refractive index. Dispersion in the polycrystalline alloy is primarily due to Fayleigh scattering above 5 mHz but there is evidence of an internal absorption process b e l o w this frequency.

The phase and group velocity spectra, calculated as above for graphite, are presented in Fig. 7

Conclusion

Our results clearly demonstrate the existence of a dispersive contribution to the prc- pagation velocity of an ultrasonic wave which can be separated into t m canponents by identifying one cmponent with the Hilbert transform of the amplitude spectrun. Processes in the material which contribute to attenuation of the ultrasonic wave will fall into t m categories: normal attenuation processes which satisfy conditions which guarantee that they are minimun- phase in nature, and other processes, such as scattering,which may contribute to the total

attenuation, an3 which may have both minimun-phase and non-minimun phase contributions to the canplex propagation constant. These terms are in addition to pure-delay contributions to the system transfer f urn tion.

Sime the minim-phase ccmponent of the propagation constant has contributions fran two distinct physical mechanisms , Kramers-Kronig relations cannot be used by thenselves to separate the contributions of each process. Furthermore, careless use of these relations may produce large errors in interpreting the role of internal absorption when scattering is present. The existence of an all-pass contribution to the phase is direct evidence of an ananolous process in absorption.

The contribution of the several mechanism to dispersion can best be seen by converting phase data to refractive index normalized to v(0). The contributions of each mechanisn to the refractive index are additive and can be seperately evaluated. Our results show that the dispersive contribution to the refractive index is closely related to the source of attenuation, the porosity of the graphite. An improved theoretical treatment of this process will illminate the connection between the volune porosity and the excess refractive index.

These results verify the importance of developing the technique of ultrasonic spectroscopy for the evaluation of materials properties and promise to provide a more comprehensive analytic tool than pulse-echo or the-of-f 1 ight methods.

In this study, we have demonstrated the feasibility of generating accurate spectra for a high loss material and, in particular, separating the total phase into its canponent parts over a wide frequency range. This technique, therefore, shows much pranise as a useful tool in studying the physics of viscoelastic media and characteriz- ing the microstructure of ceramics and composites.

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References

D. W. Fitting and L. Adler, Ultrasonic Spectral Analysis for Nondestructive Evalua- - tion, Plenm Press, NY, 1981.

E.P. Papadakis, J . Acoust. S. E. 22, 711,(1965).

J.F. Muratore, H.R. Carleton, and H. Austerlitz, Proceedings of the 1982 Ultrasonics Symposiun, San DiGo, m E , York (1982)1049.

J . F . Muratore and H.R. Carleton, 2. Acoust. Soc. supplement 1,Vo1.74 (1983)S60.

J.F. Muratore and H. R. Carleton, PrWeed- ings of the 1984 Ultrasonics Symposim, Dallas, IEEE, New York,781,(1984).

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1985 ULTRASONICS SYMPOSIUM - 1049

6 .

7.

8.

9.

10.

11.

12.

H. R. Carleton and H. Austerlitz, 2. k o u s t . SOC. pun. Supplement 1, W. 74, S77, ( 1983r -

H.W. Bode, Network Analysis and Feedback Amplifier Design, D. Van Nostrand Canpany, New York (1945).

J.S. Toll, =. e. 104, 6,(1956)1760. E.A. Robinson, Randcm Wavelets and Cykrnet ic Systems, Haf ner Pub1 ish= Cmpany, New York (1962).

Engineering Properties of Zinc Alloys, International Lead-Zinc Orqanization, 39,New York,(1981).

A.V. Cppenheim and R.W. Schafer, Digital Signal Processing, Ch. 7, Prentice-Hall, Inc., Ergled Cliffs, Ns (1975).

W. Sachse and Y.H. Pao, 2. AJ&. =. - 49(8) (1978)4320.

SIGNAL PROCESSING AND F F T POP 11/34

- - - - - - - - - - - - - - - - - -

PLOTTER

HP 214 PULSE WAVEFORM GENERATOR

1 t I 1 -

TEKTRONIX 1121 PRE-

AMPLI FlER

7

I SAMPLE I TRANSMITTER RECEIVER

F i g . 1 B lock d iagram o f u l t r a s o n i c spec t roscopy system.

FREQUENCY (VHZ)

F i g . 3 Spec t ra o f r e f r a c t i v e i n d e x components f o r g r a p h i t e : ( 1 ) a l l pass phase, (2)rninirnum phase, ( 3 ) t o t a l d i s p e r s i o n , and ( 4 ) t o t a l phase.

4000

I t 1

I

- i

i J $ 1

I

2 00 4 00 e. 00 8 00 I0 0

FREQUEhCY (MHL)

*@Be L L --- I - - - L - L 1- 1 I , - , d n 00

F i g . 4 Group ( 1 ) and phase ( 2 ) v e l o c i t y s p e c t r a f o r porous g r a p h i t e .

FREQUENCY (VHZ)

F i g . 5 P l o t of a t t e n u a t i o n c o e f f i c i e n t and l o g of a t t e n u a t i o n c o e f f i c i e n t f o r ZA-12.

F i g . 2 P l o t o f a t t e n u a t i o n c o e f f i c i e n t and l o g o f a t t e n u a t i o n c o e f f i c i e n t f o r g r a p h i t e .

1050 - 1985 ULTRASONICS SYMPOSIUM

L -_-I- a m 12. 0 16. E 28. B

-0.828 .. 80 8. 00

FREQUENCY (MHZ)

F ig . 6 Spectra of refractive index for ZA-12: ( 1 ) all pass phase, (2) minimum phase, and (3) total dispersion.

t

t t

i i

,- - L - - L - - - . L - - LA----.- 1 - - - I - . J 2.08 4 . 0 0 1. 88 a m La B

FREQUENCY (MHZ)

I a 00

F i g . 7 Group (1) and Phase (2) velocity spectra for ZA-12.

1985 ULTRASONICS SYMPOSIUM - 1051