Dispersion and frequency dependent nonlinearity

Ultrasonics 41 (2004) 709–718 www.elsevier.com/locate/ultras

Dispersion and frequency dependent nonlinearity parameters in a graphite–epoxy composite P.A. Elmore *, M.A. Breazeale Department of Physics and Astronomy, Jamie Whitten National Center for Physical Acoustics, The University of Mississippi, University, MS 38677, USA

Abstract Longitudinal phase velocity and nonlinearity parameter have been measured as a function of frequency in the low megaHertz range in a laminate graphite–fiber–epoxy–resin composite. Amplitudes of both the fundamental and generated second harmonics were measured absolutely with a capacitive receiver. Phase velocity and nonlinearity parameter vary with frequency. The extent of the variance depends on the orientation of the fiber layers. Comparison is made between the nonlinear differential equation appropriate for crystals and a new equation that accounts for frequency dependence of phase velocity and nonlinearity parameter. The newer equation describes the data more accurately than the crystalline model does, but appears to require additional terms. 2003 Elsevier B.V. All rights reserved. PACS: 43.25.Ba; 43.25.Dc; 43.35.Cg Keywords: Composites; Nonlinearity parameter; Harmonic-generation; Dispersion

1. Introduction In a nonlinear solid medium, the phase velocity of an acoustic wave is dependent on the strain of the material. As a result, the compressive and rarefied parts of an acoustic wave travel at different speeds. One part progressively catches up with the other as the wave progresses through the medium, resulting in wave steepening. This phenomenon is called harmonic generation. When the initial disturbance is a single frequency plane wave, harmonics grow with time or distance the wave travels through the medium. The applicability of the nonattenuative model with only the first nonlinear terms to solids containing grain boundaries became doubtful after the nonlinearity parameter b was measured for PZT [1] and sandstone [2]. The measured nonlinearity parameter is 1500 in PZT at the Curie temperature and is 7000 in sandstone at room temperature, whereas it is between two and fifteen for cubic crystals [3]. In addition, a frequency dependent nonlin* Corresponding author. Present address: Naval Research Laboratory, Marine Geosciences Division, Code 7440, Stennis Space Center, MS 39529. E-mail address: [email protected] (P.A. Elmore).

0041-624X/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2003.11.001

earity parameter was measured in PZT, instead of the frequency independence seen in crystals [4]. This led to theoretical model that included dispersive terms [5]. Both results indicate that the model used to describe harmonic generation in single crystals is inadequate for PZT and sandstone. A substance that may exhibit behavior similar to PZT and sandstone is composites. These solids are made from two or more materials, so numerous boundaries should appear. Fiber–epoxy composites are among the more common types manufactured because they are lightweight, yet mechanically strong. The fibers are usually glass or carbon; the resin is commonly epoxy, polyester or vinyl ester [6]. Studies of the nonlinear nature of composites have been performed by Prosser [7] and Wu and Prosser [8]. Prosser measured the effect of hydrostatic pressure and axial stress on phase velocity in a uniaxial carbon–fiber– epoxy–resin composite. Measurable increases in phase velocity were seen with applied pressure or stress, a verification of nonlinear restoring forces. In Wu and Prosser’s study, harmonic generation measurements were made for these samples at a fundamental frequency of 5 MHz. These measurements were made by using contact transducers for transmission and reception;

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P.A. Elmore, M.A. Breazeale / Ultrasonics 41 (2004) 709–718

transfer functions were used for calibration. Values were reported for b of 10.8 when sound is traveling perpendicular to the fiber axis and 4.1 when sound is traveling parallel to the axis. These values of b are within the range of nonlinearity parameters for single crystals. The frequency dependence, however, remains to be investigated. In the present study, frequency dependences of both the phase velocity and nonlinearity parameter are measured with a capacitive receiver [9]. For consistency, the samples used in this study were cut from the same bulk sample used in Prosser’s and Wu and Prosser’s studies. A theoretical treatment of the way frequency dependence enters into ultrasonic propagation is given in Section 2, followed by a description of the methods used to make measurements in Section 3. A presentation of the data is given in Section 4, followed by summary and conclusions in Section 5.

2. Theory 2.1. Frequency independent model Thurston and Shapiro [10] presented a theoretical description of nonlinear wave propagation in crystalline solids. Using Newton’s law of motion, the definition of adiabatic elastic coefficients given by Brugger [11] and the stress tensor for a solid given by Thurston [12], they derived (Einstein summation convention) o2 ui o2 uk oup our q0 2 ¼ cijkl þ Mijklpq þ Mijklpqrs þ ; ot oaj oal oaq oas ð1Þ where ui is the displacement of the ith particle when the solid is stressed, ai is the coordinate of the ith particle in its unstressed state, t is time, and cijkl is the second-order elastic modulus of the solid. The tensor Mijklpq is a linear combination of second-order and third-order elastic moduli; the Mijklpqrs include fourth-order moduli. These equations usually are difficult to solve. Simplification occurs, however, if the sound wave travels along a direction of pure mode propagation. In that case, the energy flux and displacement vectors are collinear. For crystals, these directions are given by Gerlich and Breazeale [13]. For a unidirectional composite, these directions are symmetry axes [14]. After rotation of the coordinate system to a direction of pure mode propagation, the equations of motion become ! 2 o2 u o2 u ou ou q0 2 ¼ 2 M2 þ M3 þ M4 þ ; ð2Þ ot oa oa oa where M2 , M3 , etc. are the linear combinations of elastic moduli after the coordinate transformation.

The zero-order approximation of Eq. (2) (truncation to the first term on the right-hand side) gives the linear wave equation. A solution is u ¼ A1 sinðka xtÞ with phase velocity, vph ¼ x=k. The first-order approximation to Eq. (2) has a solution u ¼ A1 sinðka xtÞ

A21 k 2 a M3 cos 2ðka xtÞ: 8 M2

ð3Þ

The solution, which can be obtained by Fourier analysis [10] or perturbation [15], allows one to define the nonlinearity parameter in terms of measured quantities b lim 8A2 =A21 k 2 a: A1 !0

ð4Þ

2.2. Frequency dependent model Van Den Abeele and Breazeale [5] accounted for frequency dependent behavior by adding terms to Eq. (2). As a result, the new nonlinear wave equation is ! 2 o2 u o2 u ou ou q0 2 ¼ 2 M2 þ M3 þ M4 þ ot oa oa oa ! 2 4 ou ou ou þ þ : þ C4 C2 þ C3 oa4 oa oa ð5Þ The new equation accounts for the frequency dependence of b observed in PZT [1] and allows for dispersion as well as frequency dependence of the nonlinearity. 2.2.1. Zero-order solution of the frequency-dependent model The zero-order approximation to Eq. (5) is o2 u o2 u o4 u ¼ M þ C : ð6Þ 2 2 ot2 oa2 oa4 Perturbation begins with a solution of the form u ¼ A1 sinðka xtÞ. Substitution of this solution into Eq. (6) yields q0

q0 x2 þ M2 k 2 C2 k 4 ¼ 0:

ð7Þ

Since k ¼ x=vph , this may be rewritten as v4ph

M2 2 4p2 C2 f 2 vph þ ¼ 0: q0 q0

ð8Þ

This equation is quadratic in v2ph . The solutions are 8 1=2 > M2 M2 16p2 q0 C2 2 > > 1 f ; < 2q0 2q0 M22 ð9a-bÞ v2ph ¼ 1=2 > M2 M2 16p2 q0 C2 2 > > : þ 1 f : 2q0 2q0 M22 It is important that these relations provide velocities that are positive real numbers. Only Eq. (9b) meets this criterion. In Eq. (9a), positive velocities are obtained only when M2 < 0. This corresponds to an unstable


equilibrium in the strain potential energy. This result is unmeaningful physically. 2.2.2. First-order solution The first-order approximation of Eq. (5) is 2 o2 u o2 u ou ou o4 u q0 2 ¼ M2 2 þ M3 þ C 2 ot oa oa2 oa oa4 4 ou ou þ C3 : 4 oa oa The solution is [5]

u ¼ A1 sinðka xtÞ 1 2 2 A k aðM3 k 2 C3 ÞðM2 8C2 k 2 Þ 8 1 cos 2ðka xtÞ ðM2 8C2 k 2 Þ2 þ 9C22 k 6 a2 3 2 A C2 k 5 a2 ðM3 k 2 C3 Þ sin 2ðka xtÞ: ð11Þ 8 1 2 ðM2 8C2 k 2 Þ þ 9C22 k 6 a2 From Eq. (11), the magnitude of the second harmonic, jA2 j is 1 2 2 A k ajM3 k 2 C3 j 8 1 : ð12Þ jA2 j ¼ 2 1=2 ½ðM2 8k 2 C2 Þ þ 9C22 k 6 a2 An equation for the magnitude of the frequency dependent nonlinearity parameter can be written as jM3 k 2 C3 j : ð13Þ jbj ¼ 2 1=2 ½ðM2 8k 2 C2 Þ þ 9C22 k 6 a2 This curve for jbj has a few features that are different from the single crystal b. The value of jbj is predicted to have a frequency dependence. Also, it can be quadratic at low frequencies or small values of C2 . When C2 and C3 vanish, Eqs. (11) and (13) reduce to the results for the crystalline model. 2.3. Dissipative model A second-order nonlinear wave equation that accounts for both harmonic generation and attenuation is given by Mendousse [16]. Using the present notation and definition of the nonlinearity parameter, this equation is (truncating to the M3 term) o2 u o3 u o2 u ou q0 2 q0 b 2 ¼ 2 M2 þ M3 ; ð14Þ ot oa ot oa oa where b is the damping parameter. This equation is reducible to Burger’s equation, given the appropriate approximations [17]. Burger’s equation is derived for solids by Norris [18]. Since Eq. (14) reduces to Burger’s equation, it should also apply to solids. The solution to Eq. (14) is [19] A2 k 2 u ¼ A0 expða1 aÞ sinðka xtÞ 0 8 expð2a1 aÞ expða2 aÞ M3 cos 2ðka xtÞ a2 2a1 M2

to first-order in ðM3 =M2 Þ, where A0 is the amplitude of the fundamental if there is no attenuation, and a1 and a2 are the attenuations at the first and second harmonic frequencies, respectively. From this solution, the definition of the nonlinearity parameter in (4) is (refined to lossy media) b ¼ lim

ð10Þ

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A1 !0

8A2 a2 2a1 : A21 k 2 1 expðða2 2a1 ÞaÞ

ð16Þ

A1 ¼ A0 expða1 aÞ is the amplitude of the fundamental where the measurement is made. (Eq. (16) reduces to Eq. (4) when the medium is lossless or the attenuation increases linearly with frequency.)

3. Experimental procedure 3.1. Sample preparation The sample is a 150-ply unidirectional laminate composite made from Thornel 300 graphite fibers and Narmco 5208 epoxy resin. The fibers are 7 ± 1 lm in diameter as measured by a calibrated microscope scale. The sample was made circa 1986 at the NASA Langley Research Center [20]. It was ultrasonically scanned at that facility for imperfections; none were reported. The density is 1.540 ± 0.005 g/cm3 . The surface resistance is approximately 30 X. The fiber volume density of the sample is assumed in Prosser’s thesis [7] to be 0.67; that assumption is maintained here. In order to use a capacitive receiver, the samples must have two parallel conductive surfaces that are optically flat. Also, the normal to the surfaces should be parallel to a symmetry axis for pure mode propagation. To locate these axes, the sample was viewed under a microscope, and photographs were taken at magnifications of ·125. A picture is shown in Fig. 1. Symmetry axes are indicated in the figure. Two samples were prepared from the bulk sample. They are shown in Fig. 2. The larger sample is used for measurements in the X and Y directions. The shorter one is used for measurements in the Z direction. The shorter sample is needed to eliminate sidewall reflections in the Z direction. A diamond wheel saw cut two parallel surfaces on opposite sides of the sample. Both sides were lapped optically flat to two wavelengths of helium light. The surfaces are parallel to 4000 , as determined from micrometer measurements. The acoustic path lengths, also determined from micrometer measurements, are listed in Table 1. 3.2. Experimental overview

ð15Þ

Ultrasound is generated by applying a high-voltage AC signal (102 Vpp in magnitude) to a lithium niobate contact transducer bonded to the surface of the sample

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Fig. 1. The carbon–fiber–epoxy–resin laminate composite sample used for this study and the orientation directions. The z-axis is parallel to the fibers. Magnification: ·125.

Fig. 2. Samples used for measurement. Ruler markings are in centimeters and millimeters.

Table 1 Acoustic path lengths in the composite samples Orientation

Path length (mm)

X Y Z

19.100 ± 0.003 20.673 ± 0.003 5.69 ± 0.01

with benzophenone or Apiezon M grease. The equipment set-up diagrammed in Fig. 3 was used to excite the transducer and measure the output from the capacitive receiver. A continuous wave (CW) signal is sent from a synthesizer/level generator to a pulsed oscillator that is acting as a gated amplifier. The frequency of the CW signal is the same as the fundamental frequency of the acoustical signal of interest, and the gated amplifier is externally triggered. Next, the signal passes through an LRC impedance bridge and a passive lowpass filter to remove harmonics from the signal. The capacitive receiver responds to both the ultrasound that passes through the sample to the bottom surface and the feed-through from the rf signal. The

Fig. 3. Experimental set-up.

output from the receiver is passed directly to a digital oscilloscope. To examine the second harmonic, the signal first goes through a passive bandpass filter, an intermediate frequency (IF) amplifier and then a 400 Hz passive highpass filter (to eliminate amplifier noise) before being displayed on the oscilloscope. To prevent aliasing, the oscilloscope was used in the interleaved sampling mode with a Nyquist frequency of 200 MHz. In the experiment, the highest frequency measured was 20 MHz in a toneburst at least 1 ls in length. As discussed in Na and Breazeale [1], the sensitivity of the capacitive receiver can be increased by placing a polyethylene film over the capacitor button. The increase in sensitivity occurs for two reasons: the dielectric property of the film increases the capacitance itself and a higher bias voltage can be used without dielectric breakdown. The uncertainty of nonlinearity parameter measurements, however, increases when the dielectric receiver is used because of uncertainty in the dielectric constant. Hence, the dielectric is used for nonlinearity parameter measurements only when signals are unobtainable otherwise. 3.3. Determining phase velocity Velocity measurements are obtained from adaptations of Papadakis’s pulse-echo-overlap [21] and videopulse [22] techniques for use with a digital oscilloscope. The individual echoes in the record can be magnified and superimposed on the computer display. The process of determining delay time is performed in the same manner as with a cathode-ray oscilloscope (CRO), except that the time between echoes now corresponds to


an offset in the computer record rather than the driving period of an oscillator. In addition to replacing the CRO with a digital oscilloscope, the delay time is determined from the time between the input signal to the transducer and the first received acoustic signal. The use of these signals minimizes the effects of diffraction in the data (when using echo-overlap techniques) and eliminates the need to consider phase shifts due to reflections at the boundary. The use of the first signal is helpful when velocity measurements are performed on highly attenuative solids like composites. The waveform used to determine phase velocity is generated and viewed on the digital oscilloscope. The length of the toneburst is adjusted so that the input signal and the acoustic signal do not overlap. A typical waveform is shown in Fig. 4a. When repetitive signals are acquired, an equally weighted average of 50–100 waveforms is performed by the digitizer to reduce the noise in the signal. When the averaging is completed, the final record is stored in the digital oscilloscope. The frequency of the signal is then lowered 5% or 10%, and the process is repeated. The second record is stored into computer memory as well and is used later to determine proper cyclic matching. For use with digital data, Papadakis’s equation for phase velocity can be modified in the following manner

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where L is the sample length, T is the delay time, n is the number of mismatched cycles, f is the frequency, sbond is the delay through the bond, dD is the diffraction correction, and dEM is the delay in the reception of the input signal due to the finite speed of light (this small quantity

could be ignored in this experiment). The quantity sbond is determined by measuring the bond thickness (or finding an upper limit) and by using a bond with a known sound speed (Benzophenone) or known acoustical impedance and density. The diffraction correction, dD is estimated from the expression given by Rogers and Van Buren [23]. The value of n for any amount of cyclic mismatch can be determined by the techniques given by McSkimin [24]. The n ¼ 0 condition occurs when the initial cycles of the input and acoustic signals overlap because there are no phase shifts from reflections or diffraction to consider. This is the condition found in this experiment. To adapt the video-pulse technique to the present situation, the waveform recorded at frequency f is copied to two other memory banks. Each of these records is offset and magnified independently. The offset between the two is adjusted until the input signal and the acoustic signal appear to overlap in the initial transient part of the waveforms with no mismatch (Fig. 4b). This offset is measured for several zero crossings. The average value is recorded. The uncertainty is a few parts in 104 . The cyclic overlap may be checked by examining the offsets with the signal record at the lower frequency. At n ¼ 0, phase change is caused only by dispersion. To adapt the echo-overlap to the present situation, the same procedure is used except that the steady state parts of the tonebursts are used rather than the transient parts (Fig. 5a and b). Some difference in the offset at the n ¼ 0 condition may be observed because diffraction corrections also must be considered with the echooverlap method. Due to the small size of the sample used for making measurements in the Z direction, a sample of neutron-

Fig. 4. Video-pulse principles. (a) The rf signal and acoustic signal at 4.75 MHz and 3 ls/div. (b) The darkened parts of (a) magnified to 150 ns/div and overlapped with correct cyclic matching.

Fig. 5. Echo-overlap principles. (a) The rf signal and acoustic signal at 4.75 MHz and 3 ls/div. (b) The darkened parts of (a) magnified to 150 ns/div and overlapped with correct cyclic matching.

vph ¼

L T n=f sbond þ dD þ dEM

;

ð17Þ

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irradiated [1 1 1] copper was used as a buffer. The transmitting transducer is bonded to one end of the buffer; the other end of the buffer is bonded to the composite sample. The delay time through both Tboth is measured and recorded. The sample is then removed, and the delay time through the buffer rod alone Tbuffer rod is measured. Corrections for the delay through the bonding material between the buffer rod and the sample and for diffraction (echo-overlap only) also are applied. Hence, the phase velocity for the Z direction is vph ¼

L ; Tboth Tbuffer rod sRodBd þ dD sample

ð18Þ

where sRodBd is the correction for the delay through the bond between the sample and the buffer rod, and dD sample is the diffraction correction for the sample. 3.4. Determining nonlinearity parameter After the phase velocities are measured, the nonlinearity parameters can be determined. The frequency of the synthesizer/level generator is set at the frequency of the fundamental. The toneburst is made 50–100 cycles long to let the oscillations created by the lowpass filter decay and provide at least 20 cycles in the steady state portion of the fundamental signal. Since the toneburst is long, the input signal interferes with the acoustic signal on the oscilloscope screen. The input signal can be eliminated from the data by recording the output when the receiver is biased and when it is grounded and, then, subtracting the two records from each other. Examples of the final fundamental and second harmonic signals are given in Fig. 6. The range of voltage levels in the signal is measured, and the median between the highest and lowest peak-to-peak voltage levels in the wave envelope of the toneburst is recorded. The amplitude and noise level in the second harmonic is measured in

Fig. 7. Extrapolation of b0 in the Y direction at 4.75 MHz. b is indicated.

the same manner. The uncertainty is 1% or less for the fundamental and between 5% and 20% for the second harmonic. Receiver calibration is performed by placing a substitutional signal into a known complex impedance. Additional details may be found in Breazeale and Philip [25] or Elmore [26]. The fundamental and second harmonic are used in jb0 ðA1 Þj ¼

8jA2 jD2L ; A21 k 2 a

ð19Þ

where DL is the diffraction correction for the fundamental. The nonlinearity parameter is found from b0 in the limit that A1 goes to zero (Fig. 7). For data analysis, it is convenient to notice that limA1 !0 b0 ðA1 Þ is approached with a horizontal tangent. In this experiment, the best value was found to be the average of the data points because a reproducible trend in b0 as A1 goes to zero was not apparent.

4. Results and analysis

Fig. 6. (a) Fundamental (4.75 MHz) and (b) second harmonic (9.50 MHz) signals for sound traveling in the X direction. The time axis is 5 ls/div.

Calibration measurements were made using single crystals of copper and germanium. The data agreed with the values reported in Hiki and Granato [27] and Bains and Breazeale [28]. Frequency dependent variations were undetectable in either measurement. When the composite sample was placed in the capacitive detector, the attenuation was found to be much greater than many single crystals. Attenuation measurements for uniaxial composites made from the same materials as our samples have been reported by Lhermitte et al. [29].


Their measurements showed that the attenuation depended linearly on frequency between 2 and 5 MHz (7.1 dB/cm at 5 MHz across the fibers). We assume similar dependence beyond 5 MHz. 4.1. Phase velocity measurements For each orientation, the echo-overlap and videopulse procedures for the digital oscilloscope were used to make measurements. Measurements were made for different frequencies. Both give the same value within the margin of error. The data obtained from the adaptation of the video-pulse procedures are plotted in Fig. 8 (these data have less uncertainty than the data from the adaptation of echo-overlap technique). Data from Prosser [7] could be plotted in the X and Y directions. In

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these directions, the phase velocity increases with frequency. The change in velocity is more gradual in the X direction (1%) than in the Y direction (3.3%). For the Z direction, the uncertainty is larger than the change in velocity for the frequencies measured. The dispersion curve for the X direction appears to extrapolate into the measurement made by Prosser at 2.25 MHz. Disagreement exists between the extrapolations and Prosser’s measurements in the Y and Z directions. A least squares analysis was performed to find the values for M2 and C2 that make Eq. (9b) fit the data. These values are given in Table 2, and the curves are the solid lines in Fig. 8; the dotted lines were used for error estimation. In Table 2, v2 is the sum of the squared differences between each data point and theory (i.e. a measure of the fit), with each difference weighted by the

Fig. 8. Dispersion for the laminate graphite–epoxy composite in the (a) X direction, (b) Y direction and (c) Z direction. The solid lines are the best fits of Eq. (9b). The dashed lines are quadratic fits to the data. Solid circles represent present measurements while the open diamonds in (a) and (b) represent measurements made by Prosser [7]. For the Z direction, Prosser reports a velocity of 8390 ± 7 m/s.

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Table 2 Least-squares results for M2 and C2 for the composite Orientation

M2 (GPa)

C2 (N)

v2

X Y Z

14.4 ± 0.5 13 ± 2 128 ± 2

)0.4 ± 0.1 )3 ± 0.5 )30 ± 5

108 890 1.3

inverse of the measurement error. The curves calculated by Eq. (9b) show increasing velocity with frequency, but the concavities of the calculated and empirical curves are opposite each other. The numbers that appear in Table 2 and the solid curves in Fig. 8 are the best possible fits of the experimental results to Eq. (9b). 4.2. Harmonic generation measurements The linear dependence of attenuation on frequency [29] leads to the conclusion that it is not necessary to account for attenuation in nonlinearity parameter measurements (cf. Eq. (16)). Using Eq. (4), measure-

ments of the absolute value of the nonlinearity parameter for the X , Y and Z directions are plotted in Fig. 9. The nonlinearity parameter of the material also was studied by Wu and Prosser [8] at 5 MHz. For reference, their measurements also are plotted in Fig. 9. Values for M3 and C3 that make Eq. (13) fit the data best were found by least-squares analysis. The values of M2 and C2 in Table 2 were used in these computations. The curve fitting was performed for the nonlinearity parameter measurements in the X , Y and Z directions of the composite, with the sign of M3 assumed negative (all solids measured to date, except fused silica, have a negative M3 ). The results are given in Table 3. There are two values for C3 given for the X and Y directions because M3 and C3 can have the same or opposite signs. The best fit of Eq. (13) to the data for the X and Y directions occurs when M3 and C3 have opposite signs. Conversely, the best fit of Eq. (13) to the data for the Z direction occurs when M3 and C3 have the same signs. Two sets of M3 and C3 are not given for the Z direction

Fig. 9. Nonlinearity parameter for the laminate graphite–epoxy composite in the (a) X direction, (b) Y direction and (c) Z direction. The solid curves are the best fits of Eq. (13). M3 and C3 have opposite signs in (a) and (b) and the same signs in (c). Solid circles represent present measurements, while the open diamond represents measurements reported by Wu and Prosser [8]. (In (b), this point partially overlaps the second data point from the vertical axis.)

P.A. Elmore, M.A. Breazeale / Ultrasonics 41 (2004) 709–718 Table 3 Least-squares results for M3 and C3 for the composite Orientation

jM3 j (GPa)

X

)265 ± 1

Y

)480 ± 5

Z

)4500 ± 50

Possible C3’s (N)

v2

300 ± 5 )1300 ± 5 16,080 ± 15 )24,570 ± 15 )66,420 ± 15

5.9 57.0 0.2 4.7 0.6

The nonitalicized numbers are the more likely of the two possible values.

because there is only one minimum in the value of v2 . Possible curves for jbj as defined by Eq. (13) were computed from these sets of M3 and C3 . The best-fit curves are shown in Fig. 9.

5. Summary and conclusions Longitudinal phase velocity and nonlinearity parameter b have been determined as a function of frequency in a carbon–fiber–epoxy–resin composite. Measurements were made along axes symmetric with the orientations of the fibers and layer boundaries. Variation with frequency is seen in phase velocity and nonlinearity parameter. Linear frequency dependence on the attenuation was observed by Lhermitte et al. [29]. From current theories, linear frequency dependence of the attenuation implies that correction for attenuation is not necessary for nonlinearity parameter measurements. Over the frequency range measured, the nonlinearity parameter decreases with frequency for all three directions studied (an exception is noted for the X direction: the nonlinearity parameter increases from 3.75 to 4.75 MHz and then decreases with frequency). The change in the nonlinearity parameter for the X and Y directions appears to be about the same for the frequency range studied. In both cases, the decrease is a factor of three. For the Z direction, the nonlinearity parameter appears to decrease more rapidly than it does in the X and Y directions. The extent of the velocity variation and the magnitudes of M3 and C3 show dependence on the propagation direction relative to the orientation of the layers. Since the nonlinear equation for crystals predicts constant nonlinearity parameter with frequency, the data are analyzed with a newer model that accounts for frequency variation. The newer model describes the nonlinearity parameter data better than the crystalline model does, although the description of the phase velocity, while better, appears to require additional terms. There are two reasons Eq. (13) appears to fit the nonlinearity data better than the dispersion relation (Eq. (9b)) fit to the velocity data. (1) As wavenumber increases, the nonlinearity parameter becomes propor-

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tional to 1=k. (Within the range of the current measurements, this is the situation for all three directions.) (2) When M3 and C3 have the same signs, the numerator is a downward curved parabola. (Numerical runs conducted for M2 different from the values reported showed that C2 , M3 and C3 adjust to new values to best fit the data, with the nonlinearity parameter curve continuing to fit the data and show the same general shape.)

Acknowledgements The Office of Naval Research (funding number N0001403WX30007), the Murata Corporation, the Jamie Whitten National Center for Physical Acoustics, and the University of Mississippi supported this research. The authors thank Drs. W.H. Prosser and J.K. Na for the composite samples, Dr. C. Ochs for the use of the photographic microscope and Drs. H.E. Bass, K. Bhatt, L.N. Bolen, and J.G. Vaughn for their helpful discussions.

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