Determination of the elastic constants of some composites by using ultrasonic velocity measurements

I~

U T T E R W O R T H E I N E M A N

Composites 26 (1995) 597 604 ,~ 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 00 l 0-4361/95/$10.00

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Determination of the elastic constants of some composites by using ultrasonic velocity measurements P.W.A. Stijnman TNO Plastics and Rubber Research Institute, PO Box 6031, 2600 JA Delft, The Netherlands (Revised 28 November 1994)

An ultrasonic immersion technique was used to determine the velocity and amplitude of sound waves in composite materials. The materials under study were E-glass/IPN, E-glass/PET, Kevlar 29/IPN, Dyneema/PE and Dyneema/Kraton. The velocity and amplitude of longitudinal and transverse sound waves at the frequency of 1 MHz were measured in three planes of symmetry at several angles of incidence. The complex stiffness matrix for the composites was calculated. (Keywords: elastic constants; ultrasonic wave measurements; stiffness matrix)

INTRODUCTION The elastic constants of orthotropic composite materials can be obtained by various different techniques. A combination of tension, shear and torsion experiments can provide the data needed for the calculation of elastic constants. More elegant methods to obtain the elastic constants are resonant methods and ultrasonic wave measurements. In this research, the ultrasonic immersion technique 1 has been used for measuring the velocity and amplitude of sound waves in composites specially designed for ballistic applications. Longitudinal and transverse sound waves are generated in the composites at oblique angles of incidence to the liquid-solid interface. To get access to all elastic constants, the composites are cut along the principal directions. The damping is calculated by performing measurements on samples having different thicknesses. The applicability of the immersion technique for the determination of elastic constants and damping will be evaluated. The immersion method has been used successfully in the literature to identify the nine elastic constants of orthotropic composites ~5 and the complex elastic constants for unidirectional lossy composites 6'7.

0 01 Q l l Q12 Q13 0 0 02 Q12 Q2= Q23 0 0 03 = QI3 Q23 Q33 0 a23 0 0 0 Q44 0 a~3 0 0 0 0 Q55 .O'12 " 0 0 0 0 0

Stiffness versus wave velocity The generalized Hooke's relation between stress and strain for a three-dimensional anisotropic material is described by a (9 × 9) stiffness matrix. For orthotropic materials, the matrix is reduced to a 6 × 6 matrix having only nine independent elastic constants s. For an orthogonal frame with axes 1, 2 and 3, the stress-strain relations are given by:

(1)

in which o.1, o'2, o-3 are the normal stress components; o'23, 0"13, 0"12 are the shear stress components; el, e2, e3 are the normal strain components; e23, e]3, ~2 are the shear strain components; and Qr.~ (r,s = 1..... 6) represent the elastic constants. In an orthotropic material three modes of propagation exist in a given direction, as defined by the unit direction vector ! (see Figure 1). Their velocities are the solutions to the Christoffel equations 9. In this research waves only propagate in planes containing two symmetry axes. As an example, consider a wave propagating in the 1,2plane. In this special case the unit direction vector ! becomes (sin 0, cos¢, 0) and the Christoffel equations read9: I(B Det. L

THEORY

0 El 0 E2 0 63 0 2e=3 0 2e13 Q66_ .2e12_

pV2phase)

~/0

0

° fl

pV2phase)

0

(C

0

(D

=0

- pV2phase)

(2)

in which Vphase is the phase velocity, p is the density and B, C, D and aft are given by9: B z Q66 c0s2 ~b -.I- Q l l sin a ,;b C = Q22 c0s2 6 + 066 s i n 2 ~b

(3)

D = Q44 cos 2 q5 + Q55 sin 2 d, O/J~ z (QI2 4- Q66) COS 4 sin 4'

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Determination of elastic constants: R W.A. Stijnman city vector *'groupdoes not point in the same direction as the phase velocity vector *'ph .... The group velocity drift angle qt is measured with respect to the unit direction vector ! and is given bye°:

3

(! Vgroup) z *'phase ~ *'phase z *'group COSlff

(5)

EXPERIMENTAL

=1 Figure 1 Definition of the orthogonal frame with axes 1, 2 and 3. and the angles ~b and 0 in the orthotropic plate

The solution of (2) is: 0 = D -- PVphas 2 e 0 = (B

(4a)

pv2hase)(C pV2hase) Ct2/32

(4b)

Equation (4a) holds for a transverse wave with particle motion along the 3-axis. The velocity of propagation of these waves is determined by the elastic constants Q44 and Q55. Equation (4b) holds for a quasi-transverse wave and a quasi-longitudinal wave, both with particle motion in the 1,2-plane. The velocities of these waves are determined by the elastic constants Q11, Q22, Qlz and Q66. When the wave direction is along the 1- or 2-axis, direct calculation of Qll and Q22 (as a result of pure longitudinal waves) and Q66 (as a result of pure transverse waves) is possible. The described procedure can also be applied for waves propagating in the 1,3- and 2,3-planes. Phase versus group velocity In the derivation presented so far, the equations for modulus are expressed in terms of phase velocity (note that *'pha~e is parallel to the unit direction vector I). Experimentally, however, one measures the group velocity. The group velocity vector points in the direction in which the wave energy actually propagates. In anisotropic media (like composites) the phase and group velocities do not coincide; they are only parallel for directions of high symmetry such as the principal directions of an orthotropic material. In general, the group velo-

The experimental set-up for this study is shown schematically in Figure 2. The pulse generator is connected to the transmitter and the oscilloscope. A synchronization pulse starts the oscilloscope clock. The transducers and the sample are positioned in a water tank. The generated sound pulse propagates through water, the sample, and again through water to the receiver. The received pulse is analysed and displayed by the oscilloscope. From the analysed signal the time of flight and the amplitude (peak/peak) are determined. Both transducers are carefully aligned with one another. A construction has been developed in which both transducers can be adjusted in the tilt and azimuth (axes of rotation). Only one of the transducers can be adjusted in the vertical and horizontal (translation) directions as well. The angle of the incident wave can be changed by rotation of the sample. Mode conversion o f waves The theory presented deals with longitudinal and transverse waves. Therefore, complex propagation of bulk, surface and plate waves has to be excluded by using a wavelength of ultrasonic sound that is sufficiently smaller than the thickness of the composite. Furthermore, the sample geometry and transducer arrangement must be taken into consideration. Since the incident medium is a liquid, transverse waves can only be produced by mode conversion upon refraction in the sample. Furthermore, the particle motion is constrained to the plane of propagation. Therefore, again considering an incident wave in the 1,2-plane, neither the waves corresponding to the solution of equation (4a) (as determined by Q44 and Qs~) nor the pure transverse wave (as determined by Q66) is generated. Q66 c a n only be determined in an indirect way by multiple measurements at

SYNC 0 S

T

~

I

U

R

:

I t--

Figure 2 Set-up for measurement of the ultrasonic velocity of sound (P, pulse generator: O, oscilloscope; T, transmitter; R, receiver; S, sample)

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COMPOSITES Volume 26 Number 8 1995

Determination of elastic constants: RW.A. Stijnman in which 0 2 is the refracted angle and Nthe drift angle. The refracted angle 0 2is calculated using equation (6) and Snell's law, and the drift angle gt is calculated according to:

oblique angles of incidence. Thus the velocities of the longitudinal wave, the quasi-longitudinal wave and the quasi-transverse wave are determined by:

1,2-plane;

•

the elastic c o n s t a n t s QII, Q22, Q12 and Q66 in the

•

the elastic constants Qll, Q33, Q13 and Q55 in the 1,3-plane; and the elastic constants Q22, Q33, Q23 and Q44 in the 2,3-plane.

•

tan gt =

By rotation of the sample with respect to the sound beam, the phase velocity can be measured as a function of the angle of propagation in the sample. A set-up with all relevant geometrical parameters is shown schematically in Figure 3. From Figure 3 it is derived that:

El Pwater

At COS 01 + At 2 1-1/2

2- -

Vwaterd

material =

1 ) i ~ ln(A//Aj)+OQ"ater(Alwater,j -Alwater'i" 2(

d2 J

d COS(02 + v/) cos ~

V

- -

/phase,/

(9)

t ----

Vphase

(ama~erialV phase/CO") + 1 V " = Vphase amateria~l V' (_0

(10)

in which ¢o is the angular frequency. (7)

Calculation of stiffness matrix First the direction vector ! has to be determined. The value is related to the refraction angle and depends on

"l

" ~

/phase0

in which A is the amplitude of the sound wave; i,j refers to measurements on different thicknesses (i Cj); N is the number of thicknesses (N > 2); awater is the constant of damping of water; Alwater,Uis a correction for the wave path through the water. From the phase velocity and the damping, the complex phase velocity, It*=lt '+iv ", is calculated according t013:

(6)

in which At = twater -- /sample (twater is the time of flight through water; tsample is the time of flight through water and sample); d is the sample thickness; Vwate~is the sound velocity in water. Values for At and 01 are obtained from the experiment; Vwater is obtained from a separate measurement. Attenuation in a composite is dominated by the viscoelastic property of the matrix ~2. Therefore the constant of damping can be determined from amplitude measurements and calculation of the phase path. Referring to Figure 3, the phase path is obtained by: /phase

(8)

in which y is the measured displacement. From amplitude measurements on the same material at the same incident angle but with different thicknesses d, the damping is obtained by averaging calculated dampings of the combined measurements:

Calculation of complex velocity

Vphase ---- ~

(y/d) c o s 0 2 - sin(02 - 01 ) (Y/d)sin02 + c°s(02 - 01)

- O2 I 14I

--

t~1

/ ["~ ~"" /p e group

A/water

r

Figure 3 Set-up for sample rotation experiments (01 = incidence angle, 02 = refraction angle, ~ = drift angle, d = thickness, y = pulse displacement, Al~ter = reduction of water path, lphase= phase path (wave), lgroup = group path (energy))

COMPOSITES Volume 26 N u m b e r 8 1995

599

Determination of elastic constants: R VV.A.Stijnman the plane of interest and the coincidence of the wave direction and the material axis at perpendicular incidence. Again consider the 1,2-plane. Now ¢ in equation (3) is solved by q) = 90 - 02 for perpendicular incidence along the 1-axis or ~ = 02 for perpendicular incidence along the 2-axis. The elastic constants Q~I and Q22follow directly from the longitudinal velocity measurements along the 1- and 2-axes:

set with its length along the second axis. Each set contained three thicknesses that were different from the thickness of the plate (see Figure 4). For these three sets. velocity measurements were performed in the 1,2-, 1,3and 2,3-planes, enabling calculation of the elastic constants in these planes.

QII = P (v~) 2

The results of the measurements are presented in Figures 5 to 9 and in Tables 2 (real part of the stiffness matrix) and 3 (imaginary part of the stiffness matrix). Vphase is obtained with a statistical accuracy of 3% (except for the Dyneema/PE composite which has an accuracy of 10%). The components of the stiffness matrix (QI1, Q22 and Q33) calculated directly from the velocity measurements have an equal accuracy, i.e. 3% (Dyneema/PE 10%). The other six components of the stiffness matrix were obtained with an accuracy of 10%. The damping measurements were not accurate. In some measurements even negative damping was calculated. An estimation for the accuracy is about 50%. For some composites the fit procedure led to negative values for the imaginary part. In these cases the imaginary part was set to zero, only the real part was fitted.

Q22 = P (v'~)2

(11)

Q12 and Q66 follow from the number of measurements N at the refracted angles 02,i (i = 1..... N): p2[l~*(t~i)]

4 --

p[v*(cb~)]2(Q~l sin2&i + Q22 cos2~bi) + Q11Q22 sin2qSi cos2&i

+ Q66(Q11 sin4qSi + Q22 c0s44~i) - p [v*(05/)2]

- (Q~2 + 2Q22QI2) sin24~i cos2~b = 0

(12)

With a multi-linear least-squares fit the elastic constants Q66 and Q12 can be calculated. This procedure is followed for the three planes of interest defined by the orthotropic 1-, 2- and 3-axes, resulting in the nine elastic constants of the stiffness matrix. All experiments were performed at a frequency of 1 MHz and at a temperature of 22 + I°C.

Materials" The following composites were investigated. •

•

• •

•

E-glass/PET Fibre: E-glass, yarn type EC17 2400, plain-weave style R24-810 9622, 810 g m 2. Matrix: poly(ethylene terephthalate) (PET) film (Hostaphan). Composite: film stacking [(0 90) / ( 45 +45)],, density 2148 kg m 3. Kevlar 29/IPN - Fibre: Kelvar 29, plain-weave style 1001. Matrix: interpenetrating network (IPN) = unsaturated polyester (UP)/isopolyurethane (isoPU) (60/40). Composite: prepreg [(0 90)],, density 1364 kg m -3. E-glass/IPN- Fibre: E-glass, plain-weave, 800 g m 2. Matrix: IPN -- UP/isoPU (60/40). Composite: prepreg [(0 90)]., density 2156 kg m 3. Dyneema/PE Fibre: Dyneema, yarn type SK66, 3/1 satin-weave style 506, 150 g m 2. Matrix: Stamylex 08026, linear low density polyethylene (LLDPE) film. Composite: film stacking [(0 90)],, density 928 kg m -3. Dyneema/Kraton Fibre: Dyneema, yarn type SK66, unidirectional, 135 g m 2. Matrix: Kraton Dl107. Composite: prepreg [090]~, density 952 kg m 3.

RESULTS

DISCUSSION AND CONCLUSIONS All composites studied have a 0/90 symmetry in the 1,2plane (except the E-glass/PET which is quasi-isotropic) and therefore (see the frame definition in Figure 1) it is expected that the Q~I and Qz2 components as well as the Qa4 and Q55 components of the stiffness matrix are identical. The data given in Table 2 show that, for most composites, Ql~ equals Q22 within 5% and that Q44 equals Q55 within 5%. The larger difference found for the Dyneema/PE composite is explained by the lower accuracy of the measurement. Furthermore, it is expected that the value of Q66 (shear in the plane containing the fibres) is not smaller than the value for Q44 and Qs5 (shear perpendicular to the plane containing the fibres). The same relation holds for QI2 compared with QJ3 and Q23. The results for the Dyneema/Kraton, Dyneema/PE and

I

I

I

All composites had a fibre volume fraction of 68+_2%, with porosity of <3%.

Sample preparation A set of plates with two or three different thicknesses was used. The 1-axis and the 2-axis correspond to the plane of the plate along the symmetry axes, and the 3axis corresponds to the thickness direction (see Figure 1). Two sets of bars were cut from the thickest sample: one set with its length along the first axis and the other

600

COMPOSITES Volume 26 Number 8 1995

2 Figure 4

Preparation of samples

Determination of elastic constants: R W.A. Stijnman E glass/IPN 4250

E >

3750

o

3250

2750

D , ....

.0""

..........

.....•

"'"..

.o"

..,." . . . . '-,,

,,,, ,0'""

-64.0

•,,

........ .

•..

,o,' "%,

",,.

,,,"'""

-32.0

.000

32.0

-•

64.0

.i}2 [o] 5 Longitudinal velocities (upper traces) and transverse velocities (lower traces) as a function of the refraction angle (02) in the E-glass/IPN composite for the three planes of interest. Experimental result (+) and fit ( ) in the 1,3-plane (02 = 0 is propagation along the 3-axis); experimental result (O) and fit ( ) in the 2,3-plane (02 = 0 is propagation along the 3-axis); experimental result (e) and fit ( ...... ) in the 1,2-plane (02 = 0 is propagation along the 1-axis) Figure

Kevlar 29/IPN 4800 2~.,,..,,.

• ............• .............• ....•

~.

4000

...,.%~,

"%

0 %

:

~-

......

~ ..."

• ".....~;

T+oo /X 2400

/ . ....

~""

"O

•."

"..

,,"

1600

/

",.,

............ ~

,

-64.0

"Ill

./ +

-32.0

"-..

~

,

.000

'.......... 32.0

64.0

6 Longitudinal velocities (upper traces) and transverse velocities (lower traces) as a function of the refraction angle (02) in the Kevlar 29/IPN composite for the three planes of interest. Symbols as in Figure 5

Figure

Dyneema/PE 4800

""

4000

~

f. . . " "

',.',"~

• . ..0........... . .• .......... . . O. . . . . . . . .~

~

'~.

O..'"

X.

132o0

e/" ..,.,

.

.

.

.: . ~....,

....

.

I.

.

.

.

.

.

.

.

.

.

.

1600

•

X

I ....... ,,-" .

-64.0

•

-32.0

•

/

J"

/

,,,)........

.

~..

j." ,,,. -

-

-

~

,

.

.~'

• ,./

.._~:__~--~_. . . . . . . . . . . . . .

~ ,

.

"..

.

I .,, :\ /

.

#.#'+ ~

.

~

/

O.

. -~-,-® ~ _ , . .

'........

~

,

.000

,

32.0

........

,

,

64.0

02 [o] 7 Longitudinal velocities (upper traces) and transverse velocities (lower traces) as a function of the refraction angle (02) in the Dyneema/PE composite for the three planes of interest. Symbols as in Figure 5 Figure

COMPOSITES Volume 26 Number 8 1995 601

Determination of elastic constants: R W.A. Stijnman

Dyneema/Kraton •

. ............ • . . . . . . . . . . . . . . .

5500 " ' "

~.............. "'".

~

...

~"

0. /

"'~" ".~

4500 •.

3500

.

.

.

.

b

2500

-64.0

-32.0

.0;0

32.0

64.0

02 [o] Figure 8 Longitudinal velocities (upper traces) and transverse velocities (lower traces) as a function of the re~action angle (~) in the Dyneem~Kraton composi~ ~ r the three planes of inte~st. Symbols as in Figure 5

E glass/PET 4000

""

........• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• .......

3500 > 3000

2500 ...@......... • ....

............. '

:

0

'.:..........................................................................2 ~:-~

•

........................

ooo -64.0

-32.0

.000

32.0

64.0

-- ~2 [°1 Figure 9 Longitudinal velocities (upper traces) and transverse velocities (lower traces) as a function of the refraction angle (02) in the E-glass/PET composite for the three planes of interest. Symbols as in Figure 5

Q66 =

Kevlar 29/IPN composites are in disagreement. It is thought that the discrepancy found for these materials is caused by the less accurate fit for the components concerned. The large difference in the magnitude of the elastic constants Qu and Q22 for the two Dyneema composites is caused by the satin weave in the PE composite. In a weave only a fraction of the fibres lies in the fibre direction (the Kraton composite has unidirectional lay-up) and, therefore, the stiffness is reduced. Comparison of both IPN composites shows a large difference in the elastic constants (except for Q11 and Q22). This large difference is caused by the deliberate introduction of poor fibre-matrix adhesion in the Kevlar 29 composite (cf. the ballistic application). The E-glass/PET is quasi-is•tropic in the 1,2-plane. Therefore the following relationship holds between the modulus, the shear modulus and the cross modulus:

602

COMPOSITES Volume 26 Number 8 1995

Qll

-- ~ 2

(13)

2

Table 1 Tensile moduli of fibres and matrices of composites investigated

Fibre E-glass Kevlar 29 Dyneema Matrix I PN PET PE ~ Kraton

Modulus (GPa)

Density (kg m 3)

72 62 92

2540 1440 970

3 4.5 0.2 1.4 x 10 3

a PE modulus refers to a compression moulded sheet

1115 1395 910 980

Determination of elastic constants: R VV.A. Stijnman

Table 2

Components of stiffness matrix (real part) for materials under study

Material

QIt (GPa)

022 (GPa)

Q33 (GPa)

Qi2 (GPa)

Q13 (GPa)

Q23 (GPa)

Q44 (GPa)

Q55 (GPa)

Q66 (GPa)

41.8 31 17.6 34 36.4

40.6 33 23.4 33 36.8

26.7 6.1 3.0 3.4 23.9

10.6 2.0 2.8 0.4 15.0

10.1 3.0 1.8 2.0 9.6

10.6 2.5 1.9 2.2 9.4

8.3 1.1 0.9 0.2 6.0

8.2 1.0 1.0 0.2 5.9

10.0 1.0 0.8 0.4 9.3

E-glass/IPN Kevlar 29/IPN Dyneema/PE Dyneema/Kraton E-glass/PET Table 3

Components of stiffness matrix (imaginary part) for materials under study

Material

Qll (GPa)

E-glass/IPN Key]at 29/IPN Dyneema/PE Dyneema/Kraton E-glass/PET

2.4 11 0.8 3 2.9

Q:2 (GPa) 1.3 5 2.6 2.6

Q33 (GPa)

Q12 (GPa)

Q13 (GPa)

Q~,3 (GPa)

Q44 (GPa)

Q55 (GPa)

Q66 (GPa)

2.4 5 0.6 0,4 3.4

1.6

1.6

0.7

1.8

0.6

1.6

0.5

0.9

0.7

0.1

2.0

1.3

1.2

2.9

0.1 2.1

0.7 2.0

Table 4 Comparison of moduli calculated using the rule of mixtures (E) with stiffness values based on the present measurements (Q)

Material E-glass/IPN E-glass/PET Kevlar 29/IPN Dyneema/PE Dyneema/Kraton

Ell (GPa)

QII (GPa)

E33 (GPa)

Q33 (GPa)

35 34 32 32 32

41 37 32 21 34

19 26 19 1.3 0.01

27 24 6 3 3

Examination of Table 2 shows an agreement with this relationship. Comparison of both E-glass composites only shows a minor difference caused by the quasi-isotropic character of the PET composite. A first approximation for the modulus in the fibre direction (the 1- or 2-direction, see Figure 1 ) can be given using the simple rule of mixtures. Perpendicular to the fibre direction (the 3-direction), a modification of the rule of mixtures based on experience is used ~4. The moduli of the used materials are listed in Table 1. An exact comparison between the rule of mixtures (modulus) and the velocity measurements (stiffness) cannot be made since the modulus has to be corrected with the Poisson's ratios (V12 , VI3 and v23) to obtain the stiffness 8. This correction leads to a multiplication factor of between 1 and 1.5 for the modulus, depending on the value for the Poisson's ratio. Furthermore, the material values listed in Table 1 are for unidirectional fibre properties. Except for the Kraton composite, weaves have been used. Therefore calculations by the rule of mixtures can only be used as estimates. A more accurate determination is obtained from (ultrasonic) measurements. Table 4 summarizes the calculated results according to the rule of mixtures and the velocity measurements in the direction of the 1 and 3 principal axes. The rule of mixtures shows good agreement with the measurements. Only the elastic constants for the Kevlar 29/IPN and the Dyneema/Kraton composites deviate considerably. The deviation of the Kevlar 29/IPN has already been discussed. The matrix in the Dyneema/Kraton composite is a thermoplastic elastomer. The modulus given in Table 1 for the Kraton holds for very small strain rates. At very high strain rates

the modulus increases considerably and the value resulting from the ultrasonic measurements is thought to hold. The damping coefficients could not be determined accurately. This is partly caused by the technique itself. The frequency used is 1 MHz. Thus, given the wave velocities (see Figures 5 to 9), the wavelength is about the same order of magnitude as the thickness of the samples. This fact might lead to dispersion effects which can explain the negative damping calculated in some measurements. Decreasing the wavelength by increasing the frequency reduces such dispersion effects. Unfortunately, frequencies about one order of magnitude higher do not penetrate through the samples, the damping increases considerably and no measurements can be performed. Therefore the frequency of 1 MHz was used. Furthermore, it was noticed that waves with a large wave path in the samples were damped considerably, so that the frequency of the penetrated wave was often reduced to 0.5 MHz. This again gives rise to distortion of the calculation of damping. Although scattering on the individual fibres is not likely to occur ~ (the fibre diameter is about 100 times smaller than the wavelength), scattering on fibre bundles in weaves may play a role since the fibre bundles are only about 10 times smaller than the wavelength. However, a larger amplitude was often measured after transmission through a thick sample compared with a thin sample. This leads to the idea that other factors may be involved, such as irregular porosity, irregular fibre volume fraction and fibre misalignment. In conclusion, the stiffness matrix of composites at 22°C and 1 MHz can be obtained rather well from ultrasonic measurements. This is a non-destructive technique and gives more information than conventional mechanical measurements. However, calculation of the imaginary part of the complex elastic constants with this set-up and these materials was less successful. ACKNOWLEDGEMENTS The author thanks Mr R.H.J.W.A. Drenth for the construction of the apparatus and Mrs C. Zoeteweij for performing the measurements. This research was possible by funding of the Dutch Ministry of Defence.

COMPOSITES Volume 26 Number 8 1995

61)3

Determination of elastic constants: RW.A. Stijnman REFERENCES 1 2 3 4 5

6 7

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Markham, M.F. Composites 1970, 2, 145 Smith, R.E. ,L Appl. Phys. 1972, 46, 2555 Hosten, B. Ultrasonics 1992, 30, 365 Kriz, R.D. and Stinchcomb, W.W. Exp. Mech. 1979, 16, 41 Pearson, L.H. and Murri, W.J. in 'Proc. Review of Progress in Quantitative NDE' (Eds D. O. Thompson and D. E. Chimenti), Plenum, New York, 1986, Vol. 5, pp. 1093-1101 Hosten, B. and Deschamps, M. J. Acoust. Soc. Am. 1987, 82, 1763 Hosten, B. J. Acoust. Soc. Am. 1991, 89, 2745

COMPOSITES Volume 26 Number 8 1995

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Chalpin, J.C. in 'Primer on Composite Materials: Analysis', Technomic, Lancaster, PA 1984, pp. 7-35 Musgrave, M.J.P. Proc. Roy. Soc. A 1954, 226, 339 Auld, B.A. 'Acoustic Fields and Waves in Solids', John Wiley and Sons, New York, 1973 Krautkrfimer, J. and Krautkr~tmer, H. in 'Werkstoffprtffung met Ultraschall', Springer, Berlin, 1961, pp. 29~,1 Kim, H.C. and Park, J.M.J. Mater. Sci. 1987, 22, 4536 Waterman, H.A. KolloidoZ. u. Z. Polymere 1963, 192, 1 Tsai, S.W. and Hahn, H.T. 'Introduction to Composite Materials', Technomic, Westport, CT, 1980