Influence of particle resonance on wave propagation in a random particulate composite

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/82/020109-06503.00/0

Voi.9(2),i09-i14,1982. Printed in the U S A Copyright (c) 1982 Pergamon Press Ltd

INFLUENCE OF PARTICLE RESONANCE ON WAVE PROPAGATION IN A RANDOM PARTICULATE COMPOSITE

V.K. Kinra, E. Ker, and S.K. Datta Department of Mechanical Engineering University of Colorado, Boulder, Colorado 80309, USA

(Received 22 August 1981; accepted for print 26 January 1982)

Introduction

Unlike their one-dimensional (plate-) and two-dimensional (fibrous-) counterparts, the three-dimensional particulate composites have received relatively little attention, particularly from the experimentalist. Recently, we have carried out measurements of the longitudinal and shear velocities, and respectively, in random particulate composites [1-3]. In [i], the attention was focused solely on the long-wavelength limit, i.e., when ka< > l , were explored. The present work focuses on the intermediate regime ka = O(i). Consider the motion of a sin$1e spherical elastic inclusion in an unbounded elastic medium in which a plane time-harmonic wave is propagating; this problem has been the subject of numerous investigations; see [6-8], for example. In [6], the scattering cross-section was calculated as a function of kla (k I and k 2 are, respectively, the longitudinal and shear wavenumbers in the ma6rix); the curve showed a number of peaks and valleys. It has been conjectured that these are due to the excitation of resonant frequencies of the ~.article. These resonance effects have been studied explicitely by Flax and ~berall [8]; they employed the resonant scattering theory of [7]. The resonances were found to occur for kla>0.5 (approximate limit). Turning now to the case of a random particulate composite, we raise the following question: To what extent does the excitation of particle resonances affect the wave propagation behavior of the composite - in particular, ? In view of the extreme mathematical complexity of the problem, an exact solution does not appear to be in sight. Here, we seek the answer in purely experimental terms. For the same constituent materials, namely, an epoxy-matrix and lead-inclusions, and for the same volume fraction of inclusions, C, two specimens were 109

ll0

V.K. KINRA, E. KER and S.K. DATTA

prepared with only the following difference: In the first, highly spherical inclusions were used; in the second the so-called lead "DUST" (to be described later) was used. The important point is that, unlike the spheres, the DUST particles are not expected to be capable of sustaining resonances; this is a purely heuristic statement. The for two cases was measurably the same when k2a<>l , however, when k2a = 0(i), was found to be significantly higher for the spherical case; the difference can only be attributed to the particle resonances.

Experimental The experimental Briefly,

procedures

have been discussed

was determined

tone-burst

(0.16
water-immersion zoelectric

the time-of-flight

[1-3] and the direct contact

the matrix material:

to 1%.

lead inclusions

spheres

of radius a = 0.66 mm;

because

it consists

sieved and particles

An epoxy

and reception.

(2) Lead "DUST".

The

(i) Lead

The lead "DUST" is so-called

amoeba-like

particles.

in the range 106-460 ~m were retained.

(about fifty) was measured under a travelling microscope, ume was assigned

Pie-

(EPON 828-Z) was used as

of two types were used:

of irregularly-shaped

Both the

[4] methods were employed.

were used for both excitation

are claimed accurate

[1-4].

of an ultrasonic

MHz) through a specimen of known thickness.

transducers

measurements

by measuring

in detail elsewhere

The dust was

A random sample

an 'estimated'

vol-

to each particle and the average volume was used to fix the

radius of an equivalent

sphere of the same volume:

erties of the constituents

a = 0.25 mm.

The prop-

are given in Table i. TABLE I

C1 (mm/~su)

C2 (mm/~su)

Density p

Poisson's Ratio, v

Epoxy

2.64

1.20

1.202

.372

Lead

2.21

0.86

Material

11.3

.411

Results For C = 5% nominal, Fig.

i.

Here and C I are, respectively,

the composite brevity,

/C I - vs. - k2a over a large range of k2a is shown in

and the matrix.

the symbols

the longitudinal

Note change of scale

(C.O.S.)

wavespeeds

in

at k2a = 5. For

SS and DS will be used to denote "spherical-specimens"

and "DUST - specimens"

respectively.

Note that when k2a < 0.75 and when

WAVE PROPAGATION IN PARTICULATE COMPOSITES

k2a>3.0 , /C I is nearly the same for SS and DS.

Iii

Thus, the difference in

the shape of DUST and spherical particles appears to make little difference in the aggregate composite properties when the wavenumber is either very small or very large. The arrow L.W.L. Datta

(long-wavelength limit) is the analytical prediction of

[5], valid when ka<
For k2a<0.75 , although the DS measurements show

excellent agreement with L.W.L., the SS measurements do not. although peculiar, is not at all disturbing,

This behavior

for in [i], where much smaller

glass spheres were used (a = 0.15.mm), the agreement between the analysis and the experiments at very long wavelengths be excellent.

(k2a as small as 0.3) was shown to

Here, the lowest available frequency was 0.16 MHz, this corres-

ponds to k2a = 0.56, which apparently is not "small enough."

According to

[6,8] the dominant resonances may occur as early as k l a = 0.5 or k 2 a = i.i. These resonances

are believed to be the reason for the

peculiar behavior noted

above. The most interesting data, however, is in the range 1.0
(approximate

Note that for exactly the same acoustical conditions

(C, k2a , etc.), for SS is significantly higher than for DS.

The differ-

ence can be attributed only to the excitation of the particle resonance for the spherical case.

Note that the largest difference is about 0.20 whereas

the measurement error is only 0.01, implying in some sense a "signal-to-noise" ratio of 20. value,

For the spherical case (SS), as k2a is increased from its low

/C 1 takes an abrupt and large (0.25) jump at about k2a = i.

After

a small "overshoot" it rapidly settles down to what appears to be its high frequency limit of 1.0.

On the other hand, for the DUST - specimen (DS) as

k2a is increased, /C 1 increases gradually in the range l
Recall that from [6,8] the range in which

dominant resonances occur is roughly kla>0.5 or k2a>l.l.

It is not at all

clear why the resonance effects do not cause a difference in for DS and SS for k2a>3.

Finally, for large k2a both the SS and the DS measurements

agree remarkably well with the results of a short-wave length analysis (k2a>>l) by Datta

[2] in which lead sphears are treated like an acoustically

equivalent non-viscous fluid. In order to insure that the results obtained in Fig. 1 were not fortuitous, the experiments were repeated with a higher volume fraction: C = 15% (nominal.

112

V.K. KINRA, E. KER and S.K. DATTA

Essentially the same results were obtained, these are shown in Fig. 2. following features are noteworthy:

(i)

In Fig. i, SS case, for k 2 a < 0.75,

/C 1 was found to be significantly below its L.W.L; somewhat peculiar.

It is reassuring,

therefore,

the same phenomenon is observed once again; /C 1 occurs at k2aZl for both C;

this observation was

to note that for C = 15%

(2) For the SS case, the jump in

(3) Significant increase in C from 5%

to 15% results in only small and subtle changes in the curves; "overshoot" for the SS in is seen for both cases; DS case.

The

(4)

The

it is absent in the

Finally, the numerical values of data presented in Figures 1 and 2

are given in Table 2.

Conclusion

When k2a = 0(i), the wave propagation behavior of a random particulate

com-

posite containing spherical inclusions is significantly influenced by the excitation of the particle resonances.

Acknowledgement

The financial support of the National Science Foundation under the grant ENG78-10168 to the University of Colorado is gratefully acknowledged.

References

i.

V.K. Kinra, M.S. Petraitis and S.K. Datta, Int. J. Solids and Structures, Vol. 16, p. 301 (1980).

2.

V.K. Kinra and S.K. Datta,

3.

V.K. Kinra and A. Anand, Int. J. Solids and Structures

4.

E. Ker, M.S. Thesis, Department of Mechanical Engineering, University of Calorado, Boulder, Colorado. (1981).

5.

S.K. Datta, in Continuum Model of Discrete Systems, Ed. by J.W. Provan, p. iii, University of Waterloo Press, (1978).

6.

R. Truell, C. Elbaum, and B.B. Chick, Ultrasonic Methods in Solid State Physics, Academic Press, New York, (1969).

7.

L. Flax, L.R. Dragonette, and H. Uberall, J. Acout. p. 723, (1978).

8.

L. Flax and H. Uberall, J. Acoust. Soc. Am., Vol. 67, No. 5, p. 1432, (1980).

(to be communicated). (in press).

Soc. Am., Vol. 63,

WAVE PROPAGATION IN PARTICULATE COMPOSITES

113

TABLE 2 "C'= 5% a=0.25 mm

k2a

/C I

~ = 15% a=0.66 ram (continued) k2a

/C 1

a=O. 25 mm

k2a

! /C I

a=O. 66 mm

k2a

/C 1

0.65

0.85

2.25

1.01

0.39

0.71

0.59

0.65

1.31

0.87

2.25

1.02

0.43

0.70

0.80

0.62

1.31

0.86

2.42

1.01

0.48

0.71

1.04

0.78

1.77

0.85

2.59

1.01

0.52

0.73

1.14

0.80

2.21

0.93

2.59

1.01

0.59

0.72

1.28

O. 89

2.62

0.97

2.76

i. O0

0.65

0.71

1.38

0.96

3.14

0.99

2.76

1.02

0.66

0.70

1.38

0.93

6.55

1.01

2.94

1.02

0.72

0.70

1.73

0.99

3.11

1.02

0.78

0.70

2.07

1.02

3.28

1.01

0.85

0.69

2.42

1.04

a--O. 66 mm 0.56

0.78

3.46

1.01

0.92

0.69

2.42

1.05

0.69

0.81

3.63

1.00

0.98

0.70

2.76

1.04

1.28

1.06

3.80

1.01

1.05

0.69

3.11

1.02

1.31

1.07

3.97

1.01

1.97

0.77

3.46

1.02

1.38

1.06

4.15

1.01

2.62

0.81

3.80

1.02

1.55

1.03

4.32

1.01

3.28

0.93

4.15

1.01

1.73

1.01

4.49

1.01

4.59

0.98

4.50

1.02

1.73

1.02

4.49

1.00

4.84

1.00

1.90

1.01

4.67

1.01

5.18

1.01

2.07

1.02

4.84

1.01

5.88

0.97

5.01

1.01

7.95

1.00

5.88

1.00

12. i0

1.00

7.95

1.00

13.82

i. O0

12. i0

i. O0

13.82

1.00

114

V.K.

I

KINRA,

E. KER and S.K. DATTA

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k# FIG.

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11 13 15

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N o r m a l i z e d longitudinal wave v e l o c i t y vs. shear w a v e n u m b e r C.O.S. = Change of Scale; L.W.L. = l o n g - w a v e l e n g t h limit [5].

~c~

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q

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1.0

•

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•

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•

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! C=15"Io

0.9

o •

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0.8

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DUST SPHERE

THEORY ~2]

®

0.7

O--

o

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0.61 0

"I

1

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11 13

~a FIG.

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N o r m a l i z e d longitudinal wave v e l o c i t y vs. shear w a v e n u m b e r C.O.S. = Change of Scale; L.W.L. = l o n g - w a v e l e n g t h limit [5].

15