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Microwave excitation of ultrasound in graphite–fiber reinforced composite plates
Ultrasonics 41 (2003) 97–103 www.elsevier.com/locate/ultras
Microwave excitation of ultrasound in graphite–fiber reinforced composite plates Emmanuel Guilliorit a, Bernard Hosten a, Christophe Bacon a b
a,*
, D.E. Chimenti
b
Laboratoire de M ecanique Physique, Universit e Bordeaux 1, UMR 5469 CNRS 351, cours de la Lib eration, 33405 Talence Cedex, France Aerospace Engineering and Engineering Mechanics Department, and Center for NDE, Iowa State University, Ames, IA 50011-2271, USA Received 10 April 2002; accepted 9 October 2002
Abstract In this paper is demonstrated the effect of microwave beam polarization on the thermal generation of acoustic waves in continuous fiber-reinforced composite laminates. It is found that beam polarization strongly influences the dielectric interaction that leads to thermal losses, bulk expansion, and acoustic wave generation. The oriented graphite fibers in the composite laminate effectively short the microwave fields and reduce the generation efficiency nearly to zero. Ultrasonic waves at several hundred kHz generated in the composite are detected by air-coupled acoustic transducers located on the opposite side of the plate specimen from the 9.41 GHz incident microwave beam. With some averaging signal-to-noise ratios of better than 26 dB are obtained. Applying a conventional model of electromagnetic wave scattering in anisotropic media to this experiment yields good agreement between calculations and measured data. Implications for microwave-acoustic testing of graphite-reinforced composites are also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic waves; Electromagnetic microwaves; Uniaxial graphite–epoxy composite; Nondestructive inspection
1. Introduction The thermal excitation of acoustic waves by an incident, time-varying microwave beam has been demonstrated recently in a series of experiments [1,2] designed to explore the dependence of microwave interaction with elastic vibrations and propagating acoustic waves in matter. Several interesting results have now been established as a consequence of these earlier investigations. In slightly absorbing media there will be, in general, sufficient microwave power absorbed to lead to observable differential thermal expansion of the sample. An earlier simplified theoretical model proposed by the authors assumed that the spatial distribution of the temperature increase caused by the microwave energy absorption was either uniform, linear [2,3] (this assumption is nearly valid for weakly absorbing materials), or exponential [4,5]. In recent measurements a 9.41 GHz waveguide has been filled with a bar of PVC a few centimeters to tens of centimeters in length during high-
*
Corresponding author. Tel.: +33-5-57-96-22-72. E-mail address: [email protected] (C. Bacon).
power pulsed microwave excitation. The ensuing electromagnetic absorption and thermal expansion leads to standing acoustic waves in the bar [6] whose reinforcement is strongly dependent on acoustic wavelength and on the bar length. The explanation of the resulting acoustic standing-wave resonances in terms of absorbed microwave power and physical properties of the bar has proven very accurate and can account for essentially all the observed phenomena, including a much higherfrequency resonance resulting from the finite microwave wavelength at 10 GHz. These results have been reported and form the basis of the current investigation. In the current work we seek to demonstrate and explain the differential absorption, and therefore generation of acoustic power, in samples of uniaxial graphite– epoxy composite. The graphite–epoxy is particularly well suited to this extension of the original work, because it has a highly anisotropic electrical conductivity. The electrical behavior of graphite-reinforced composites has been fully investigated by others [7,8] in the context of spectroscopy and for the purpose of nondestructive evaluation [9,10]. In these earlier works it has been found that the conductivity of graphite fibers is fully sufficient to support induced currents and further that the highly
0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 2 ) 0 0 4 3 2 - 8
98
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
directional nature of fiber placement in a typical composite laminate leads to a very anisotropic conductivity, or more generally, dielectric tensor. It was this realization that encouraged us to investigate further the extraordinary properties of fiber reinforced composites in the context of the microwave-acoustic work. We have found entirely predictable behavior of the composite laminate in experimental measurements designed to expose the biaxial behavior of the material in microwaveacoustic coupling. In modeling calculations we have treated the composite laminate as a highly anisotropic complex dielectric medium, whose dielectric tensor is a reflection of the structural anisotropy of the composite plate. The agreement between the simple model and the experiments is quite good and completely expected on the basis of the earlier eddy current results. The plan of this paper is as follows: Section 2 summarizes the calculation of electromagnetic scattering of a biaxially anisotropic dielectric plate in a vacuum, Section 3 briefly reviews the experimental procedure and Section 4 presents detailed experimental results, along with comparisons to the model calculations. The paper ends with a summary of the salient conclusions.
2. Theoretical summary The calculation of scattering properties of an anisotropic slab has been investigated by a number of researchers for a variety of problems connected with electromagnetic wave propagation [11,12]. In this article we draw on the work of Bagatskaya et al. [13] to analyze the scattering coefficients for an anisotropic slab with a complex dielectric constant. These authors treat a much more general case than the classical problem but with a relatively simple approach that preserves the clarity of the solution. To summarize briefly, they begin with a fully anisotropic dielectric tensor, which we have particularized to biaxial symmetry, that is, where two of three diagonal tensor elements are degenerated. The wave is assumed to propagate in the z direction with a wavenumber k ¼ kz , and the optic axis is chosen as the X axis, as shown in Fig. 1. The X direction has a complex valued dielectric tensor element eXX ¼ ek , and the Y and z directions have identical values of the dielectric tensor element, eYY ¼ ezz ¼ e? . The two axes x and y are chosen to be parallel to the incident electric field E i and to the incident magnetic field H i , respectively. The angle that the incident, linearly polarized electromagnetic wave E i makes with the optic axis is denoted by u. A slab of thickness h lies in the XY plane with one surface at the origin, and a plane electromagnetic wave is incident upon this surface. MaxwellÕs equations in the absence of free charges can be expressed conveniently using the BerremanÕs
Fig. 1. Geometry of the problem.
formalism that leads naturally to an application of the Hamilton–Cayley theorem. Several earlier authors have also taken this approach [12,14]. For a single layer, such as we are dealing with here, the procedure is much simpler than the general multilayered calculation given in [13]. The final matrix of variables connecting the electromagnetic field components on either side of the slab is the result of this portion of the calculation. From this matrix and the boundary conditions on the fields at the slab surfaces, the scattering coefficients––reflection and transmission––can be constructed. Quoting a simplified form of these relations appropriate for our geometry in this problem, the permittivity tensor of the uniaxial dielectric medium in the chosen coordinate system x, y, z can be expressed as 0 1 e1 e4 0 eu ¼ @ e4 e2 0 A ð1Þ 0 0 e3 with tensor elements defined by 2 e1 ¼ ek cos2 u þ e? sin2 u; 6 6 e2 ¼ ek sin2 u þ e? cos2 u; 6 4 e3 ¼ e? ; e4 ¼ ðe? ek Þ cos u sin u:
ð2Þ
Considering a plane monochromatic wave that propagates in a linear medium whose optical properties depend on the single Cartesian coordinate z, we can cast MaxwellÕs equations in the Berreman form dWðzÞ ¼ ixAWðzÞ; dz
ð3Þ
where x is the angular frequency, and WðzÞ is a column matrix containing the amplitudes of the transverses fields
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
0
1 Ex ðzÞ B Ey ðzÞ C C WðzÞ ¼ B @ Hx ðzÞ A; Hy ðzÞ and the Berreman matrix A is given by 0 1 0 0 0 l0 B 0 0 l0 0 C C: A¼B @ e4 e2 0 0 A e1 e4 0 0
ð4Þ
ð5Þ
From Eq. (3) the transverse components of the electromagnetic field on the upper boundary Wð0Þ are related to the components on the lower boundary WðhÞ by WðhÞ ¼ P Wð0Þ;
Then, Eq. (6) can be written 0 1 1 Exi þ Exr Ext B C B Eyt C Eyr B C: B C i r A @ Y t Et A ¼ P @ Y Ey y t t i i r Y Ex Y ðEx Ex Þ 0
Y t 0 ðP41 Y i P44 Þ ðP42 þ Y i P43 Þ 0 1 P11 þ Y i P14 B P þ Y iP C 24 C B 21 ð11Þ B C: @ P31 þ Y i P34 A
ryx
ð6Þ
ð7Þ
To proceed we do not use the Hamilton–Cayley theorem to calculate the matrix exponential expðixhAÞ. Instead, we compute the P matrix numerically. For an electromagnetic wave linearly polarized along the x axis, the Frenel formulas are 2 r Ex ¼ rxx Exi ; 6 Eyr ¼ ryx Exi ; 6 ð8Þ 4 Et ¼ txx Ei ; x x Eyt ¼ tyx Exi ; where Er and Et are respectively the amplitudes of the reflected wave and the transmitted wave. In this system, the complex coefficients of reflection rxx and transmission txx describe the transformation of an incident wave linearly polarized along the x axis into a wave of the same polarization, whereas the two coefficients ryx and tyx describe the transformation of a linearly polarized wave into a wave of orthogonal polarization. In the case of an isotropic medium, MaxwellÕs equations lead to relations among components of the electromagnetic field 2 kzt t t t t 6 Hx ¼ xl0 Ey ¼ Y Ey 6 4 kzt t Hyt ¼ þ xl E ¼ þY t Ext 0 x and 2 kzi r Hxr ¼ þ xl E ¼ þY i Eyr 6 0 y : 4 kzi r Hyr ¼ xl Ex ¼ Y i Exr 0
ð10Þ
With the use of relation (8), and having divided both sides of Eq. (8) by Exi , the four complex coefficients of transmission and reflection are given by 0 1 0 11 txx 1 0 ðP11 Y i P14 Þ ðP12 þ Y i P13 Þ Bt C B 0 1 ðP21 Y i P24 Þ ðP22 þ Y i P23 Þ C B yx C B C B C¼B C @ rxx A @ 0 Y t ðP31 Y i P34 Þ ðP32 þ Y i P33 Þ A
where the transfer matrix P is defined by P ¼ expðixhAÞ:
99
P41 þ Y i P44 In this relation Pjk (j, k ¼ 1, 2, 3, 4) are the elements of the transfer matrix P . If the incident wave is monochromatic at a frequency f0 , the average electromagnetic power per unit surface is given by 1 P ¼ ReðE HÞ ez ; 2
ð12Þ
where ez is an unit vector along the z axis and H is the complex conjugate of H. On the upper side of the plate (z ¼ 0), the average power is i 1 h P ð0Þ ¼ Re ðExi þ Exr Þð Hyi þ Hyr Þ Eyr Hxr : ð13Þ 2 If we introduce Eq. (9), and if we suppose that the plate is in vacuum (Y i ¼ Y t ¼ 1=l0 c real), we finally obtain for the average power on the upper side (c is the speed of light) P ð0Þ ¼
Exi2 2 2 ð1 jrxx j jryx j Þ: 2l0 c
ð14Þ
On the lower side where z ¼ h, a similar calculation leads to P ðhÞ ¼
Exi2 2 2 ðjtxx j þ jtyx j Þ: 2l0 c
ð15Þ
i
ð9Þ
The quantities Y t and Y i are the inverses of the wave impedance in the transmitted and in the incident medium.
The mean power P transported by the incident wave is calculated by writing rxx ¼ 0 and ryx ¼ 0 in Eq. (14). We obtain i
P ¼
Exi2 : 2l0 c
ð16Þ a
With this assumption the mean power P absorbed by the dielectric plate is given by
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a
P ¼ P ð0Þ P ðhÞ i
¼ P ½1 jrxx ðx0 Þj2 jtxx ðx0 Þj2 jryx ðx0 Þj2 jtyx ðx0 Þj2 : ð17Þ
For short microwave pulses where heat conduction can be neglected, the heat equation is oh oP ¼ ; ð18Þ ot oz where h is the temperature rise, and C the specific heat. Because the thickness of the dielectric plate is very small (only a few millimeters), we make the assumption that the electromagnetic power decreases linearly inside the plate. Consequently, the spatial distribution of the temperature rise caused by the microwave energy absorption should be uniform with the axial coordinate z.Then, by using Eq. (18), the temperature rise after one period T0 (T0 is generally very short compared to the pulse width s) is given by qC
a
hðz; tÞ ¼
hðz; T0 Þ P sqðtÞ; sqðtÞ ¼ T0 qCh
where qðtÞ is 2 0 qðtÞ ¼ 4 t=s 1
ð19Þ
a function defined by for for for
t 6 0; 0 6 t 6 s; t P s:
ð20Þ
In the Fourier domain the acceleration a~ðh; xÞ of the acoustic waves generated at z ¼ h by the sudden thermal expansion is (the detailed calculation are given in Ref. [5]) H12 G2 H22 G1 a~ðh; xÞ ¼ ; ð21Þ H12 where x is the angular frequency of the acoustic waves. The components of the matrix H and the vector G are defined by q ! c s xS H¼ ; ð22Þ xS q s c 0 1 qb ~ h ðc 1Þ 2 A: ð23Þ G ¼ @S bx S h~s In these relations c ¼ cosðxS hÞ and s ¼ sinðxS hÞ, h~ðxÞ is the Fourier transform of the temperature rise, b is the expansion coefficient, and S is the complex slowness. We will use these expressions to model the data collected in the experiment described in the following section.
3. Experimental procedure Fig. 2 demonstrates the measurement geometry of the experiment. Time-gated microwaves are produced by a magnetron at 9.41 GHz with a peak power at 5.5 kW.
Fig. 2. Experimental set-up.
The maximum pulse width is 1 ls, and the repetition rate of several hundred kHz is set using a function generator. This configuration produces a burst limited typically to a total of 50–100 pulses. Finally, the burst is repeated such that the time average of the on–off ratio of the microwave generator is limited to less than 0.001, corresponding to the generatorÕs continuous power limit. This condition leads to a repetition rate of the burst of about 1 kHz. The electromagnetic waves are delivered by a standard waveguide that permits only the fundamental waveguide mode TE10 to propagate. To obtain the maximum power, a hybrid ‘‘tee’’ impedance adapter is inserted into the circuit and adjusted to achieve a minimum voltage standing wave ratio (VSWR), as measured by a wattmeter. At the end of the waveguide the radiation is coupled to free space by the waveguide flange itself. The radiation is allowed to shine on a 3-mm thick piece of graphite–epoxy composite laminate standing within 10 mm of the waveguide flange end. On the opposite side of the specimen, as shown in Fig. 2, an air-coupled transducer is placed at about 150 mm from the plate to monitor its microwave-induced elastic vibrations. Measurements have been performed on a single uniaxial composite plate. The sample is oriented so that its planar surface is normal to the microwave Poynting vector. Then, rotating the sample in the microwave field produces a variation in the angle that the linearly polarized wave makes with the direction of the continuous fiber reinforcements. Absorbed microwave power through dielectric relaxation in the polymer matrix of the composite then accounts for the generation, by means of thermal coupling, of a stress wave at the frequency of the microwave burst repetition rate in the composite laminate. This stress wave couples to the surrounding air, as would occur in any vibrating plate, and these pressure oscillations in the air are detected by the specially designed air-coupled ultrasonic transducer. The procedure used here follows closely the one established
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101
for such measurements in our earlier work [2–6]. The stress wave in the composite can be enhanced, as we showed earlier, by choosing a repetition frequency of the microwave pulses in the burst that corresponds to a mechanical resonance of the plate. The amplitude of the generated stress wave, monitored by the air-coupled probe (essentially a high-frequency microphone), is recorded as a function of the sample orientation angle. An example of the unprocessed data is shown in Fig. 3. In the upper frame Fig. 3(a) is a representation of the microwave power output, illustrating the finite train of bursts. The lower frame, Fig. 3(b), is raw data of the response monitored by the air-coupled probe as collected on a LeCroy digital oscilloscope, averaged 64 times. The microwave Poynting vector in this case is oriented normal to the composite fibers, and the electric field is perpendicular to the composite fibers (u ¼ 90°).
4. Results and discussion The results of this study are relatively straightforward, and we present them simply as the received acoustic wave amplitude versus fiber orientation angle. The first sample studied is a uniaxial graphite–epoxy laminate (AS/4-3501) consisting of 24 plies, and the measurements and calculations are presented in Fig. 4. The error bars for each measurement correspond to the dispersion obtained for three measurements. Fig. 4 represents the amplitudes of the acoustic signals as a function of the orientation angle. From this figure we see that the acoustic amplitude is measured to be the largest at an orientation angle of 90° and 270°, that is, when the electric field is polarized normal to the fibers. The amplitude is smallest for angles of 0° and 180°. The plot of Fig. 4 mirrors exactly the symmetry of the laminate itself, as we would entirely expect. Furthermore, the calculated transmission coefficient magnitude Txx ¼ jtxx j for the microwave field amplitude has the same symmetry and angular behavior as the acoustic wave detected by the air-coupled transducer, as we can
Fig. 3. (a) Microwave power output and (b) temporal acoustic response (u ¼ 90°).
Fig. 4. Experimental and theoretical amplitudes.
see in Fig. 5(a). We believe the reason for this is quite clear. When the microwave electric field is oriented
Fig. 5. Moduli of the coefficient of transmission of (a) Txx ¼ jtxx j and Tyx ¼ jtyx j and (b) Rxx ¼ jrxx j and Ryx ¼ jryx j versus u.
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E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
normal to the graphite fibers, there will be little influence of the graphite conductivity on the effective composite dielectric constant. The effective medium permittivity will be a weakly complex number with a real part near unity and a much smaller imaginary part. This is the parameter identified as e? in Section 2. On the other hand, ek describes the effective medium dielectric constant along the fiber direction. In this case, the fiber conductivity is substantial, and we have modeled this constant with a value of ek ¼ 3 103 i. The large imaginary part of the permittivity insures a substantial induced current in the fibers. This induced current, in turn, shields the interior of the laminate from the microwave fields and thus little power is deposited in the sample interior, where losses lead to thermal heating and subsequently to excitation of stress waves in the laminate. Instead, most of the power is simply reflected and the stress-wave amplitude drops nearly to zero. The moduli of the coefficient of reflection is given in Fig. 5(b). For all calculations reported here the other parameters are e? ¼ 3 i , f0 ¼ 9:41 GHz and h ¼ 3 mm. The model accounts for this behavior as well, and its angular dependence follows closely that of the measurements. It should be noted at this point that model calculation developed by Bagatskaya et al. [13] assumes, in satisfying superficial continuity conditions, that the dielectric properties of the medium never approach those of a conductor. In this sense we are pushing the model beyond the point of its intended application, because the fiber dielectric properties are more conductor-like than insulator-like.
5. Conclusions We have shown in this paper how a microwave beam interacts with a carbon-fiber reinforced composite laminate to produce a thermally generated ultrasonic wave in the laminate. As the polarization of the microwave beam is rotated from normal to parallel to the uniaxial continuous fibers in the composite, the microwave energy available to be absorbed in the dielectric matrix decreases dramatically, resulting in a corresponding reduction in the amplitude of the generated acoustic wave. The major fraction of the microwave energy is reflected back towards the source by the conducting fibers. The model of microwave scattering of Bagatskaya et al. [13] adequately accounts for these effects, if we assume a bidirectional composite dielectric constant, normal and parallel to the fibers. The circumstance of high microwave reflectivity of the graphite fibers suggests that inspection methods based on this effect of thermally generated acoustic waves will have to account for this finding. The results of Figs. 4 and 5 suggest superficially that there may still be transmission of microwave energy into the sample
interior as long as no ply lamina have fibers oriented in the 0° or 180° directions. For a single layer this result certainly holds, as indicated experimentally and theoretically in Fig. 4. Once the microwave beam has passed through a 45° layer, however, the beam character changes in important ways. The fibers act as a polarizer, and the microwaves emerging from an interaction with a 45° ply (as shown in Fig. 4) do indeed have an amplitude that can be surmised from the figure. Their polarization, however, will no longer have the same orientation as the incident beam. Instead, all transmitted microwave energy will now be polarized along the fiber direction, that is, along 45°. Then, the interaction with a second ply having an orientation of 135°, or 90° from the first ply (as in a biaxial composite), will effectively block all further propagation of microwaves into the composite. The reason for this phenomenon is the polarizing effect of the first layer that effectively shorts all fields parallel to the graphite fibers. The implication for nondestructive inspection is that only laminates with graphite fibers concentrated in a single direction or near a single axis would be candidates for inspection using a technique based on this thermalexpansion effect. Clearly, in the uniaxial case the method functions well, as we have shown here. In the biaxial case, however, the first ply layer shorts the only polarization that would propagate through the second layer. Therefore, the bulk of the microwave energy is reflected and does not reach the sample interior, where it could have generated elastic waves. Quasi-uniaxial lay-ups might also be inspected using this technique, where fibers have a ply lay-up of [0,þ20,)20]. In this case, as in the uniaxial case, there should be sufficient energy transmitted into the sample to excite elastic waves. The effect might also be employed as the basis for a sensitive inspection method to find misaligned plies or wavy graphite fibers, both serious lay-up defects. The preliminary results reported here indicate the possible utility and promise of this approach. References [1] B. Hosten, P.A. Bernard, Ultrasonic wave generation by timegated microwaves, J. Acoust. Soc. Am. 104 (2) (1998) 860–866. [2] C. Bacon, B. Hosten, P.A. Bernard, Acoustic wave generation in viscoelastic rods by time-gated microwaves, J. Acoust. Soc. Am. 106 (1) (1999) 195–201. [3] E. Guilliorit, C. Bacon, B. Hosten, Generation dÕondes acoustiques par micro-ondes impulsionnelles, 14eme Congres Francßais de Mecanique, paper 91 (on cd-rom), Toulouse, France (1999). [4] B. Hosten, C. Bacon, Measurement of complex young moduli of composite materials by time-gated microwaves, in: D.O. Thompson, D.E. Chimenti (Eds.), Proceedings of Review of Progress in Quantitative Non-Destructive Evaluation, vol. 19, Plenum Press, New York, 2000. [5] C. Bacon, B. Hosten, E. Guilliorit, One-dimensional prediction of the acoustic waves generated in a multilayer viscoelastic body by microwave irradiation, J. Sound Vibrat. 238 (5) (2000) 853–867.
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103 [6] C. Bacon, E. Guilliorit, B. Hosten, D. Chimenti, Acoustic waves generated by pulsed microwaves in viscoelastic rods: modeling and experimental verification, J. Acoust. Soc. Am. 110 (3) (2001) 1398–1407. [7] Y. Chekanov, R. Ohnogi, S. Asai, M. Sumita, Electrical properties of epoxy resin filled with carbon fibers, J. Mater. Sci. 34 (22) (1999) 5589–5592. [8] S. Wang, D.D.L. Chung, Electrical behavior of carbon fibers polymer matrix composites in the through-thickness direction, J. Mater. Sci. 35 (1) (2000) 91–100. [9] M.H. Choi, R. Ohnogi, B.H. Jeon, I.J. Chung, The effect of coupling agent on electrical and mechanical properties of carbon fiber/phenolic resin composites, Polymer 41 (2000) 3243–3252. [10] P. Boinard, R.A. Pethricki, W.M. Banks, R.L. Crane, Nondestructive evaluation of adhesively bonded composite structures
[11]
[12] [13]
[14]
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using high frequency dielectric spectroscopy, J. Mater. Sci. 35 (6) (2000) 1331–1337. N.P. Zhuck, Electromagnetic theory of arbitrarily anisotropic layered media Part 1. Scalarisation of fields quantities, Int. J. Electron. 75 (1) (1993) 141–148. C. Oldano, Electromagnetic-wave propagation in anisotropic stratified media, Phys. Rev. A 40 (10) (1989) 6014–6020. O.V. Bagatskaya, N.P. Zhuck, E.B. Malets, S.N. ShulÕga, Analytical calculation of the complex reflection and transmission coefficients of a dielectric plate with allowance for the electromagnetic anisotropy of the material, Russ. J. Nondestruct. Test. 33 (1) (1997) 63–74. M. Becchi, P. Galatoa, Berreman-matrix formulation of light propagation in stratified anisotropic chiral media, Eur. Phys. J. B 35 (1999) 399–404.
Microwave excitation of ultrasound in graphite–fiber reinforced composite plates Emmanuel Guilliorit a, Bernard Hosten a, Christophe Bacon a b
a,*
, D.E. Chimenti
b
Laboratoire de M ecanique Physique, Universit e Bordeaux 1, UMR 5469 CNRS 351, cours de la Lib eration, 33405 Talence Cedex, France Aerospace Engineering and Engineering Mechanics Department, and Center for NDE, Iowa State University, Ames, IA 50011-2271, USA Received 10 April 2002; accepted 9 October 2002
Abstract In this paper is demonstrated the effect of microwave beam polarization on the thermal generation of acoustic waves in continuous fiber-reinforced composite laminates. It is found that beam polarization strongly influences the dielectric interaction that leads to thermal losses, bulk expansion, and acoustic wave generation. The oriented graphite fibers in the composite laminate effectively short the microwave fields and reduce the generation efficiency nearly to zero. Ultrasonic waves at several hundred kHz generated in the composite are detected by air-coupled acoustic transducers located on the opposite side of the plate specimen from the 9.41 GHz incident microwave beam. With some averaging signal-to-noise ratios of better than 26 dB are obtained. Applying a conventional model of electromagnetic wave scattering in anisotropic media to this experiment yields good agreement between calculations and measured data. Implications for microwave-acoustic testing of graphite-reinforced composites are also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic waves; Electromagnetic microwaves; Uniaxial graphite–epoxy composite; Nondestructive inspection
1. Introduction The thermal excitation of acoustic waves by an incident, time-varying microwave beam has been demonstrated recently in a series of experiments [1,2] designed to explore the dependence of microwave interaction with elastic vibrations and propagating acoustic waves in matter. Several interesting results have now been established as a consequence of these earlier investigations. In slightly absorbing media there will be, in general, sufficient microwave power absorbed to lead to observable differential thermal expansion of the sample. An earlier simplified theoretical model proposed by the authors assumed that the spatial distribution of the temperature increase caused by the microwave energy absorption was either uniform, linear [2,3] (this assumption is nearly valid for weakly absorbing materials), or exponential [4,5]. In recent measurements a 9.41 GHz waveguide has been filled with a bar of PVC a few centimeters to tens of centimeters in length during high-
*
Corresponding author. Tel.: +33-5-57-96-22-72. E-mail address: [email protected] (C. Bacon).
power pulsed microwave excitation. The ensuing electromagnetic absorption and thermal expansion leads to standing acoustic waves in the bar [6] whose reinforcement is strongly dependent on acoustic wavelength and on the bar length. The explanation of the resulting acoustic standing-wave resonances in terms of absorbed microwave power and physical properties of the bar has proven very accurate and can account for essentially all the observed phenomena, including a much higherfrequency resonance resulting from the finite microwave wavelength at 10 GHz. These results have been reported and form the basis of the current investigation. In the current work we seek to demonstrate and explain the differential absorption, and therefore generation of acoustic power, in samples of uniaxial graphite– epoxy composite. The graphite–epoxy is particularly well suited to this extension of the original work, because it has a highly anisotropic electrical conductivity. The electrical behavior of graphite-reinforced composites has been fully investigated by others [7,8] in the context of spectroscopy and for the purpose of nondestructive evaluation [9,10]. In these earlier works it has been found that the conductivity of graphite fibers is fully sufficient to support induced currents and further that the highly
0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 2 ) 0 0 4 3 2 - 8
98
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
directional nature of fiber placement in a typical composite laminate leads to a very anisotropic conductivity, or more generally, dielectric tensor. It was this realization that encouraged us to investigate further the extraordinary properties of fiber reinforced composites in the context of the microwave-acoustic work. We have found entirely predictable behavior of the composite laminate in experimental measurements designed to expose the biaxial behavior of the material in microwaveacoustic coupling. In modeling calculations we have treated the composite laminate as a highly anisotropic complex dielectric medium, whose dielectric tensor is a reflection of the structural anisotropy of the composite plate. The agreement between the simple model and the experiments is quite good and completely expected on the basis of the earlier eddy current results. The plan of this paper is as follows: Section 2 summarizes the calculation of electromagnetic scattering of a biaxially anisotropic dielectric plate in a vacuum, Section 3 briefly reviews the experimental procedure and Section 4 presents detailed experimental results, along with comparisons to the model calculations. The paper ends with a summary of the salient conclusions.
2. Theoretical summary The calculation of scattering properties of an anisotropic slab has been investigated by a number of researchers for a variety of problems connected with electromagnetic wave propagation [11,12]. In this article we draw on the work of Bagatskaya et al. [13] to analyze the scattering coefficients for an anisotropic slab with a complex dielectric constant. These authors treat a much more general case than the classical problem but with a relatively simple approach that preserves the clarity of the solution. To summarize briefly, they begin with a fully anisotropic dielectric tensor, which we have particularized to biaxial symmetry, that is, where two of three diagonal tensor elements are degenerated. The wave is assumed to propagate in the z direction with a wavenumber k ¼ kz , and the optic axis is chosen as the X axis, as shown in Fig. 1. The X direction has a complex valued dielectric tensor element eXX ¼ ek , and the Y and z directions have identical values of the dielectric tensor element, eYY ¼ ezz ¼ e? . The two axes x and y are chosen to be parallel to the incident electric field E i and to the incident magnetic field H i , respectively. The angle that the incident, linearly polarized electromagnetic wave E i makes with the optic axis is denoted by u. A slab of thickness h lies in the XY plane with one surface at the origin, and a plane electromagnetic wave is incident upon this surface. MaxwellÕs equations in the absence of free charges can be expressed conveniently using the BerremanÕs
Fig. 1. Geometry of the problem.
formalism that leads naturally to an application of the Hamilton–Cayley theorem. Several earlier authors have also taken this approach [12,14]. For a single layer, such as we are dealing with here, the procedure is much simpler than the general multilayered calculation given in [13]. The final matrix of variables connecting the electromagnetic field components on either side of the slab is the result of this portion of the calculation. From this matrix and the boundary conditions on the fields at the slab surfaces, the scattering coefficients––reflection and transmission––can be constructed. Quoting a simplified form of these relations appropriate for our geometry in this problem, the permittivity tensor of the uniaxial dielectric medium in the chosen coordinate system x, y, z can be expressed as 0 1 e1 e4 0 eu ¼ @ e4 e2 0 A ð1Þ 0 0 e3 with tensor elements defined by 2 e1 ¼ ek cos2 u þ e? sin2 u; 6 6 e2 ¼ ek sin2 u þ e? cos2 u; 6 4 e3 ¼ e? ; e4 ¼ ðe? ek Þ cos u sin u:
ð2Þ
Considering a plane monochromatic wave that propagates in a linear medium whose optical properties depend on the single Cartesian coordinate z, we can cast MaxwellÕs equations in the Berreman form dWðzÞ ¼ ixAWðzÞ; dz
ð3Þ
where x is the angular frequency, and WðzÞ is a column matrix containing the amplitudes of the transverses fields
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
0
1 Ex ðzÞ B Ey ðzÞ C C WðzÞ ¼ B @ Hx ðzÞ A; Hy ðzÞ and the Berreman matrix A is given by 0 1 0 0 0 l0 B 0 0 l0 0 C C: A¼B @ e4 e2 0 0 A e1 e4 0 0
ð4Þ
ð5Þ
From Eq. (3) the transverse components of the electromagnetic field on the upper boundary Wð0Þ are related to the components on the lower boundary WðhÞ by WðhÞ ¼ P Wð0Þ;
Then, Eq. (6) can be written 0 1 1 Exi þ Exr Ext B C B Eyt C Eyr B C: B C i r A @ Y t Et A ¼ P @ Y Ey y t t i i r Y Ex Y ðEx Ex Þ 0
Y t 0 ðP41 Y i P44 Þ ðP42 þ Y i P43 Þ 0 1 P11 þ Y i P14 B P þ Y iP C 24 C B 21 ð11Þ B C: @ P31 þ Y i P34 A
ryx
ð6Þ
ð7Þ
To proceed we do not use the Hamilton–Cayley theorem to calculate the matrix exponential expðixhAÞ. Instead, we compute the P matrix numerically. For an electromagnetic wave linearly polarized along the x axis, the Frenel formulas are 2 r Ex ¼ rxx Exi ; 6 Eyr ¼ ryx Exi ; 6 ð8Þ 4 Et ¼ txx Ei ; x x Eyt ¼ tyx Exi ; where Er and Et are respectively the amplitudes of the reflected wave and the transmitted wave. In this system, the complex coefficients of reflection rxx and transmission txx describe the transformation of an incident wave linearly polarized along the x axis into a wave of the same polarization, whereas the two coefficients ryx and tyx describe the transformation of a linearly polarized wave into a wave of orthogonal polarization. In the case of an isotropic medium, MaxwellÕs equations lead to relations among components of the electromagnetic field 2 kzt t t t t 6 Hx ¼ xl0 Ey ¼ Y Ey 6 4 kzt t Hyt ¼ þ xl E ¼ þY t Ext 0 x and 2 kzi r Hxr ¼ þ xl E ¼ þY i Eyr 6 0 y : 4 kzi r Hyr ¼ xl Ex ¼ Y i Exr 0
ð10Þ
With the use of relation (8), and having divided both sides of Eq. (8) by Exi , the four complex coefficients of transmission and reflection are given by 0 1 0 11 txx 1 0 ðP11 Y i P14 Þ ðP12 þ Y i P13 Þ Bt C B 0 1 ðP21 Y i P24 Þ ðP22 þ Y i P23 Þ C B yx C B C B C¼B C @ rxx A @ 0 Y t ðP31 Y i P34 Þ ðP32 þ Y i P33 Þ A
where the transfer matrix P is defined by P ¼ expðixhAÞ:
99
P41 þ Y i P44 In this relation Pjk (j, k ¼ 1, 2, 3, 4) are the elements of the transfer matrix P . If the incident wave is monochromatic at a frequency f0 , the average electromagnetic power per unit surface is given by 1 P ¼ ReðE HÞ ez ; 2
ð12Þ
where ez is an unit vector along the z axis and H is the complex conjugate of H. On the upper side of the plate (z ¼ 0), the average power is i 1 h P ð0Þ ¼ Re ðExi þ Exr Þð Hyi þ Hyr Þ Eyr Hxr : ð13Þ 2 If we introduce Eq. (9), and if we suppose that the plate is in vacuum (Y i ¼ Y t ¼ 1=l0 c real), we finally obtain for the average power on the upper side (c is the speed of light) P ð0Þ ¼
Exi2 2 2 ð1 jrxx j jryx j Þ: 2l0 c
ð14Þ
On the lower side where z ¼ h, a similar calculation leads to P ðhÞ ¼
Exi2 2 2 ðjtxx j þ jtyx j Þ: 2l0 c
ð15Þ
i
ð9Þ
The quantities Y t and Y i are the inverses of the wave impedance in the transmitted and in the incident medium.
The mean power P transported by the incident wave is calculated by writing rxx ¼ 0 and ryx ¼ 0 in Eq. (14). We obtain i
P ¼
Exi2 : 2l0 c
ð16Þ a
With this assumption the mean power P absorbed by the dielectric plate is given by
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a
P ¼ P ð0Þ P ðhÞ i
¼ P ½1 jrxx ðx0 Þj2 jtxx ðx0 Þj2 jryx ðx0 Þj2 jtyx ðx0 Þj2 : ð17Þ
For short microwave pulses where heat conduction can be neglected, the heat equation is oh oP ¼ ; ð18Þ ot oz where h is the temperature rise, and C the specific heat. Because the thickness of the dielectric plate is very small (only a few millimeters), we make the assumption that the electromagnetic power decreases linearly inside the plate. Consequently, the spatial distribution of the temperature rise caused by the microwave energy absorption should be uniform with the axial coordinate z.Then, by using Eq. (18), the temperature rise after one period T0 (T0 is generally very short compared to the pulse width s) is given by qC
a
hðz; tÞ ¼
hðz; T0 Þ P sqðtÞ; sqðtÞ ¼ T0 qCh
where qðtÞ is 2 0 qðtÞ ¼ 4 t=s 1
ð19Þ
a function defined by for for for
t 6 0; 0 6 t 6 s; t P s:
ð20Þ
In the Fourier domain the acceleration a~ðh; xÞ of the acoustic waves generated at z ¼ h by the sudden thermal expansion is (the detailed calculation are given in Ref. [5]) H12 G2 H22 G1 a~ðh; xÞ ¼ ; ð21Þ H12 where x is the angular frequency of the acoustic waves. The components of the matrix H and the vector G are defined by q ! c s xS H¼ ; ð22Þ xS q s c 0 1 qb ~ h ðc 1Þ 2 A: ð23Þ G ¼ @S bx S h~s In these relations c ¼ cosðxS hÞ and s ¼ sinðxS hÞ, h~ðxÞ is the Fourier transform of the temperature rise, b is the expansion coefficient, and S is the complex slowness. We will use these expressions to model the data collected in the experiment described in the following section.
3. Experimental procedure Fig. 2 demonstrates the measurement geometry of the experiment. Time-gated microwaves are produced by a magnetron at 9.41 GHz with a peak power at 5.5 kW.
Fig. 2. Experimental set-up.
The maximum pulse width is 1 ls, and the repetition rate of several hundred kHz is set using a function generator. This configuration produces a burst limited typically to a total of 50–100 pulses. Finally, the burst is repeated such that the time average of the on–off ratio of the microwave generator is limited to less than 0.001, corresponding to the generatorÕs continuous power limit. This condition leads to a repetition rate of the burst of about 1 kHz. The electromagnetic waves are delivered by a standard waveguide that permits only the fundamental waveguide mode TE10 to propagate. To obtain the maximum power, a hybrid ‘‘tee’’ impedance adapter is inserted into the circuit and adjusted to achieve a minimum voltage standing wave ratio (VSWR), as measured by a wattmeter. At the end of the waveguide the radiation is coupled to free space by the waveguide flange itself. The radiation is allowed to shine on a 3-mm thick piece of graphite–epoxy composite laminate standing within 10 mm of the waveguide flange end. On the opposite side of the specimen, as shown in Fig. 2, an air-coupled transducer is placed at about 150 mm from the plate to monitor its microwave-induced elastic vibrations. Measurements have been performed on a single uniaxial composite plate. The sample is oriented so that its planar surface is normal to the microwave Poynting vector. Then, rotating the sample in the microwave field produces a variation in the angle that the linearly polarized wave makes with the direction of the continuous fiber reinforcements. Absorbed microwave power through dielectric relaxation in the polymer matrix of the composite then accounts for the generation, by means of thermal coupling, of a stress wave at the frequency of the microwave burst repetition rate in the composite laminate. This stress wave couples to the surrounding air, as would occur in any vibrating plate, and these pressure oscillations in the air are detected by the specially designed air-coupled ultrasonic transducer. The procedure used here follows closely the one established
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for such measurements in our earlier work [2–6]. The stress wave in the composite can be enhanced, as we showed earlier, by choosing a repetition frequency of the microwave pulses in the burst that corresponds to a mechanical resonance of the plate. The amplitude of the generated stress wave, monitored by the air-coupled probe (essentially a high-frequency microphone), is recorded as a function of the sample orientation angle. An example of the unprocessed data is shown in Fig. 3. In the upper frame Fig. 3(a) is a representation of the microwave power output, illustrating the finite train of bursts. The lower frame, Fig. 3(b), is raw data of the response monitored by the air-coupled probe as collected on a LeCroy digital oscilloscope, averaged 64 times. The microwave Poynting vector in this case is oriented normal to the composite fibers, and the electric field is perpendicular to the composite fibers (u ¼ 90°).
4. Results and discussion The results of this study are relatively straightforward, and we present them simply as the received acoustic wave amplitude versus fiber orientation angle. The first sample studied is a uniaxial graphite–epoxy laminate (AS/4-3501) consisting of 24 plies, and the measurements and calculations are presented in Fig. 4. The error bars for each measurement correspond to the dispersion obtained for three measurements. Fig. 4 represents the amplitudes of the acoustic signals as a function of the orientation angle. From this figure we see that the acoustic amplitude is measured to be the largest at an orientation angle of 90° and 270°, that is, when the electric field is polarized normal to the fibers. The amplitude is smallest for angles of 0° and 180°. The plot of Fig. 4 mirrors exactly the symmetry of the laminate itself, as we would entirely expect. Furthermore, the calculated transmission coefficient magnitude Txx ¼ jtxx j for the microwave field amplitude has the same symmetry and angular behavior as the acoustic wave detected by the air-coupled transducer, as we can
Fig. 3. (a) Microwave power output and (b) temporal acoustic response (u ¼ 90°).
Fig. 4. Experimental and theoretical amplitudes.
see in Fig. 5(a). We believe the reason for this is quite clear. When the microwave electric field is oriented
Fig. 5. Moduli of the coefficient of transmission of (a) Txx ¼ jtxx j and Tyx ¼ jtyx j and (b) Rxx ¼ jrxx j and Ryx ¼ jryx j versus u.
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E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103
normal to the graphite fibers, there will be little influence of the graphite conductivity on the effective composite dielectric constant. The effective medium permittivity will be a weakly complex number with a real part near unity and a much smaller imaginary part. This is the parameter identified as e? in Section 2. On the other hand, ek describes the effective medium dielectric constant along the fiber direction. In this case, the fiber conductivity is substantial, and we have modeled this constant with a value of ek ¼ 3 103 i. The large imaginary part of the permittivity insures a substantial induced current in the fibers. This induced current, in turn, shields the interior of the laminate from the microwave fields and thus little power is deposited in the sample interior, where losses lead to thermal heating and subsequently to excitation of stress waves in the laminate. Instead, most of the power is simply reflected and the stress-wave amplitude drops nearly to zero. The moduli of the coefficient of reflection is given in Fig. 5(b). For all calculations reported here the other parameters are e? ¼ 3 i , f0 ¼ 9:41 GHz and h ¼ 3 mm. The model accounts for this behavior as well, and its angular dependence follows closely that of the measurements. It should be noted at this point that model calculation developed by Bagatskaya et al. [13] assumes, in satisfying superficial continuity conditions, that the dielectric properties of the medium never approach those of a conductor. In this sense we are pushing the model beyond the point of its intended application, because the fiber dielectric properties are more conductor-like than insulator-like.
5. Conclusions We have shown in this paper how a microwave beam interacts with a carbon-fiber reinforced composite laminate to produce a thermally generated ultrasonic wave in the laminate. As the polarization of the microwave beam is rotated from normal to parallel to the uniaxial continuous fibers in the composite, the microwave energy available to be absorbed in the dielectric matrix decreases dramatically, resulting in a corresponding reduction in the amplitude of the generated acoustic wave. The major fraction of the microwave energy is reflected back towards the source by the conducting fibers. The model of microwave scattering of Bagatskaya et al. [13] adequately accounts for these effects, if we assume a bidirectional composite dielectric constant, normal and parallel to the fibers. The circumstance of high microwave reflectivity of the graphite fibers suggests that inspection methods based on this effect of thermally generated acoustic waves will have to account for this finding. The results of Figs. 4 and 5 suggest superficially that there may still be transmission of microwave energy into the sample
interior as long as no ply lamina have fibers oriented in the 0° or 180° directions. For a single layer this result certainly holds, as indicated experimentally and theoretically in Fig. 4. Once the microwave beam has passed through a 45° layer, however, the beam character changes in important ways. The fibers act as a polarizer, and the microwaves emerging from an interaction with a 45° ply (as shown in Fig. 4) do indeed have an amplitude that can be surmised from the figure. Their polarization, however, will no longer have the same orientation as the incident beam. Instead, all transmitted microwave energy will now be polarized along the fiber direction, that is, along 45°. Then, the interaction with a second ply having an orientation of 135°, or 90° from the first ply (as in a biaxial composite), will effectively block all further propagation of microwaves into the composite. The reason for this phenomenon is the polarizing effect of the first layer that effectively shorts all fields parallel to the graphite fibers. The implication for nondestructive inspection is that only laminates with graphite fibers concentrated in a single direction or near a single axis would be candidates for inspection using a technique based on this thermalexpansion effect. Clearly, in the uniaxial case the method functions well, as we have shown here. In the biaxial case, however, the first ply layer shorts the only polarization that would propagate through the second layer. Therefore, the bulk of the microwave energy is reflected and does not reach the sample interior, where it could have generated elastic waves. Quasi-uniaxial lay-ups might also be inspected using this technique, where fibers have a ply lay-up of [0,þ20,)20]. In this case, as in the uniaxial case, there should be sufficient energy transmitted into the sample to excite elastic waves. The effect might also be employed as the basis for a sensitive inspection method to find misaligned plies or wavy graphite fibers, both serious lay-up defects. The preliminary results reported here indicate the possible utility and promise of this approach. References [1] B. Hosten, P.A. Bernard, Ultrasonic wave generation by timegated microwaves, J. Acoust. Soc. Am. 104 (2) (1998) 860–866. [2] C. Bacon, B. Hosten, P.A. Bernard, Acoustic wave generation in viscoelastic rods by time-gated microwaves, J. Acoust. Soc. Am. 106 (1) (1999) 195–201. [3] E. Guilliorit, C. Bacon, B. Hosten, Generation dÕondes acoustiques par micro-ondes impulsionnelles, 14eme Congres Francßais de Mecanique, paper 91 (on cd-rom), Toulouse, France (1999). [4] B. Hosten, C. Bacon, Measurement of complex young moduli of composite materials by time-gated microwaves, in: D.O. Thompson, D.E. Chimenti (Eds.), Proceedings of Review of Progress in Quantitative Non-Destructive Evaluation, vol. 19, Plenum Press, New York, 2000. [5] C. Bacon, B. Hosten, E. Guilliorit, One-dimensional prediction of the acoustic waves generated in a multilayer viscoelastic body by microwave irradiation, J. Sound Vibrat. 238 (5) (2000) 853–867.
E. Guilliorit et al. / Ultrasonics 41 (2003) 97–103 [6] C. Bacon, E. Guilliorit, B. Hosten, D. Chimenti, Acoustic waves generated by pulsed microwaves in viscoelastic rods: modeling and experimental verification, J. Acoust. Soc. Am. 110 (3) (2001) 1398–1407. [7] Y. Chekanov, R. Ohnogi, S. Asai, M. Sumita, Electrical properties of epoxy resin filled with carbon fibers, J. Mater. Sci. 34 (22) (1999) 5589–5592. [8] S. Wang, D.D.L. Chung, Electrical behavior of carbon fibers polymer matrix composites in the through-thickness direction, J. Mater. Sci. 35 (1) (2000) 91–100. [9] M.H. Choi, R. Ohnogi, B.H. Jeon, I.J. Chung, The effect of coupling agent on electrical and mechanical properties of carbon fiber/phenolic resin composites, Polymer 41 (2000) 3243–3252. [10] P. Boinard, R.A. Pethricki, W.M. Banks, R.L. Crane, Nondestructive evaluation of adhesively bonded composite structures
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