Group velocity of cylindrical guided waves in anisotropic laminate composites Evgeny Glushkov,a) Natalia Glushkova, and Artem Eremin Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, 350040, Russia
Rolf Lammering Institute of Mechanics, Helmut-Schmidt-University/University of the Federal Armed Forces, Hamburg, D-22043, Germany
(Received 25 June 2013; revised 21 October 2013; accepted 28 October 2013) An explicit expression for the group velocity of wave packets, propagating in a laminate anisotropic composite plate in prescribed directions, is proposed. It is based on the cylindrical guided wave asymptotics derived from the path integral representation for wave fields generated in the composites by given localized sources. The expression derived is theoretically confirmed by the comparison with a known representation for the group velocity vector of a plane guided wave. Then it is experimentally validated against laser vibrometer measurements of guided wave packets generated by a piezoelectric wafer active sensor in a composite plate. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4829534] V PACS number(s): 43.35.Cg, 43.35.Pt, 43.20.Rz, 43.35.Zc [JBL]
I. INTRODUCTION
The study of elastic guided wave (GW) propagation in layered structures is important for such applications as geophysics, seismology, ultrasonic non-destructive testing, structural health monitoring (SHM), and design of piezoelectrically based microelectronic devices on surface acoustic waves. In SHM, damage detection in laminate plate-like structures is based on GW generation and sensing by piezoelectric wafer active sensors (PWASs) attached to the inspected structure. The GWs, scattered by a local structural inhomogeneity, reveal the presence of a flaw, while its location is estimated from the time of signal arrival (the time of flight—TOF) registered by the elements of a sensor network. The size and form of the flaw may be assessed through its imaging using various techniques, e.g., by the embedded ultrasonic structural radar.1 Most of the imaging algorithms rely on the TOF of transmitted and scattered signals, which depend on the distance and GW group velocities in the frequency band used.2 With isotropic structures those algorithms work well, since the wavenumbers, velocities, and dispersion dependencies for any direction of wave propagation are the same. With anisotropic composite plates, the algorithms of flaw detection are also based on the TOF estimation for plane guided waves (PGWs). However, here their characteristics are directionally dependent. In general, the directivity of the group velocity vector cg of a plane wave does not coincide with the directivity of the wave vector k,3 and the direction of wave packet propagation, specified by the vector cg , is not orthogonal to the wavefront associated with the wave vector k. Hence, for a GW packet propagating in the observation direction required, one has to tune the plane
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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Pages: 148–154
wave orientation before the group velocity calculation. The PGW directivity is frequency dependent4–6 that creates additional complications for the TOF evaluation and makes computer modeling much more time consuming. On the other hand, there is no necessity to use plane waves for the TOF evaluation in SHM, since the waves generated by comparatively small PWAS transducers or scattered by localized obstacles are cylindrical guided waves (CGWs). Moreover, the asymptotic expressions derived for CGWs generated by a given force source7 allow one to avoid those difficulties, as they provide a straightforward algorithm for the group velocity calculation in any prescribed direction. These asymptotics were numerically tested and experimentally validated in the course of investigation of frequency dependent CGW directivity8 and reconstruction of effective elastic moduli9 in composite plates. The present paper is focused on a theoretical and experimental validation of the CGW group velocity calculation based on these asymptotic representations. First, we repeat a summary of these expressions, while the exhaustive formulas and algorithms are available in Ref. 7. Then, we show that these formulas yield the same steering angles and group velocity magnitudes as the ones obtained for plane waves. Finally, wave number and group velocity dispersion curves for the fundamental A0 and S0 modes propagating from the source in various fixed directions are tested against experimental data obtained based on laser vibrometer scanning of PWAS generated out-of-plane GW amplitudes. II. CYLINDRICAL GUIDED WAVES
GW propagation caused in a laminate composite plate by a localized force source is described by the solution of a three-dimensional (3D) boundary value problem for the elastodynamic equations in displacements Cijkl ul;jk þ qx2 ui ¼ 0;
0001-4966/2014/135(1)/148/7/$30.00
i ¼ 1; 2; 3;
(1)
C 2014 Acoustical Society of America V
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given in each sublayer Dm of a multilayer domain D ¼ [M m¼1 Dm : jxj < 1; jyj < 1; H z 0 (Fig. 1) with specific values of the elastic stiffness tensor components Cijkl and material density q. The displacement vector u ¼ fu1 ; u2 ; u3 g is the complex amplitude of the steady-state harmonic oscillation ueixt with the angular frequency x ¼ 2pf ; f is frequency. Hereinafter, we try to keep the previous notations.7 The sublayers Dm are perfectly bonded with each other at the interfaces z ¼ zm , m ¼ 2; 3; :::; M, while the outer sides z ¼ z1 ¼ 0 and z ¼ zMþ1 ¼ H are stress-free except in an area X, where a surface load qðx; yÞeixt , simulating a source action (e.g., PWAS), is given. The response of the structure on the applied load is derived in terms of the convolution of its Green’s matrix kðx; xÞ with the vector-function q: ðð kðx n; y g; zÞqðn; gÞdndg: (2) uðx; xÞ ¼ X
The columns kj of the 3 3 Green’s matrix k ¼ ðk1 ⯗k2 ⯗k3 Þ are the displacement vectors corresponding to concentrated point loads applied at the origin O along the coordinate vectors ij , j ¼ 1; 2; 3. Alternatively, Eq. (2) may be written as
an inverse Fourier transform of the product of the Fourier symbols K ¼ F xy ½k and Q ¼ F xy ½q: uðx; xÞ ¼ F 1 xy ½KQ ð ð 1 Kða1 ; a2 ; zÞQða1 ; a2 Þ ¼ ð2pÞ2 C1 C2 eiða1 xþa2 yÞ da1 da2 :
(3)
The integration paths C1 and C2 go along the real axes deviating from them into the complex planes a1 and a2 for rounding real poles fn of the matrix K elements. We intentionally use the notations k and K for the Green’s matrices of layered structures and their Fourier symbols, to differ them from the classical matrices of fundamental solutions in the whole space. It coincides with the traditional wavenumber notation k ¼ jkj, but it should not be confusing for they appear in different contexts. If the space variables x ¼ ðx1 ; x2 ; x3 Þ ¼ ðx; y; zÞ and the Fourier parameters a ¼ ða1 ; a2 Þ are taken in the cylindrical and polar coordinates ðr; u; zÞ and ða; cÞ [see Fig. 1(b)]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ r cos u 2 þ y2 r ¼ pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ a cos c 2 2 ; y ¼ r sin u; a2 ¼ a sin c a ¼ a1 þ a2 0 c; u 2p; z¼z
(4)
then integral representation (3) takes the form ð ð 2p 1 Kða; c; zÞQða; cÞeiarcosðcuÞ dcada; uðxÞ ¼ ð2pÞ2 Cþ 0 (5) where Cþ is an integration contour, going in the complex plane a along the real semi-axis Re a 0, Im a ¼ 0 also bypassing real poles fn . These poles are roots of the characteristic equation Dða1 ; a2 ; xÞ ¼ Dða; c; xÞ ¼ 0;
(6)
where D is the denominator of the matrix K elements. To a constant factor, it coincides with the determinant of the matrix A resulting from the substitution of general solutions of Eq. (1) in the Fourier transform domain into the transformed boundary conditions at the surfaces and interfaces z ¼ zm , m ¼ 1; 2; ::::; M þ 1 [see Eqs. (38) and (39) in Ref. 7]. The application of the residual theorem and the stationary phase method to the integrals over a and c, respectively, leads to a far-field expansion in terms of CGWs. This expansion may be obtained in various forms. More precise formulas follow from the representations in terms of cylindrical Hankel functions Hmð1Þ ðfn rÞ [e.g., from Eq. (20) in Ref. 7], while a more physically evident one is uðxÞ ¼
N X un ðxÞ þ OððfrÞ1 Þ; fr ! 1; n¼1
FIG. 1. (Color online) Sketch of the problem: (a) Side view and (b) top view and coordinate systems. J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
un ðxÞ ¼
Mn X pffiffiffiffiffi anm ðu; zÞeisnm ðuÞr = fr:
(7)
m¼1
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Here N is the number of real poles fn ðc; xÞ, and f is a characteristic wavenumber. Every term un is associated with one generating pole fn , while, generally, it may consist of Mn > 1 terms describing CGWs with the amplitude factors anm and the wavenumbers snm : snm ðuÞ ¼ sn ðbnm Þ; sn ðbÞ ¼ fn ðhÞsinb; h ¼ b þ u þ p=2:
(8)
The number Mn of wavenumbers is determined by the number of roots bnm of the stationary phase equation fn ðhÞ cos b þ f0n ðhÞ sin b ¼ 0;
(9)
where f0n is the derivative of fn with respect to its angular argument. In the special case of isotropic materials, fn are angular independent, hence, f0n ¼ 0, Mn ¼ 1, bn1 ¼ p=2, and sn1 ¼ fn . The amplitude factors anm are expressed via the residues of K elements from the poles fn and the values of the Fourier symbol Qða; cÞ at the points a ¼ snm and c ¼ u: anm ¼ bnm ðu; zÞQðsnm ; uÞ; bnm ¼ jn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if=ð2ps00n ðbnm ÞÞ fn res Kða; hnm ; zÞja¼fn : (10)
The coefficient jn is nearly always jn ¼ 1, except in the case of infrequent irregular real poles yielding backward modes for which jn ¼ 1. Since the angular dependent values snm ðuÞ play the role of CGW wavenumbers, the group velocities of wave envelopes propagating in the u direction should be cg ðuÞ ¼ ½dsnm =dx1 :
are spectral points of the homogeneous problem under consideration. For any fixed angle c, they link k and x, specifying dispersion properties of the corresponding plane wave modes. Non-zero eigensolutions an ðzÞ, associated with these roots, are cross-sectional eigenforms of the related PGWs. Since the characteristic equation remains the same, the wavenumbers kn ðc; xÞ coincide with the poles fn ðc; xÞ in the CGW asymptotics above. It is worth noting again that the wavenumbers of the latter are not fn but snm of form (8). Let x ¼ WðkÞ ¼ Wðk; cÞ be one of the roots in the frequency domain. (To simplify notations, the subscript n is usually omitted.) The slowness, phase, and group velocity vectors of the related PGW are defined via W and k as4–6 s ¼ k=W;
cp ¼ ðW=k2 Þk
and
cg ¼ grad WðkÞ:
The vectors s and cp are collinear with k, while the group velocity cg may deviate from k. Being the gradient of WðkÞ, the vector cg is orthogonal to the wave-vector (or slowness) curve jkðcÞj, which is drawn in the wavenumber plane ðkx ; ky Þ by the tip of vector kðcÞ as c varies in the range 0 c 2p [Fig. 2(a)]. With an anisotropic sample, this curve ceases to be a simple circle, so that the polar angle wðcÞ of the vector cg ¼ cg ðcosw; sinwÞ, cg ¼ jcg j, generally differs from c. The angle # ¼ wðcÞ c of the deviation between cg and k is referred to as a steering angle.4–6 Reciprocally, the vector kðcÞ is orthogonal to the closed curve formed by the tip of vector cg ðwÞ, 0 w 2p [a group velocity curve, Fig. 2(b)]. Physically, it is centrally similar to the wavefront
(11)
Note that after the roots bnm are obtained from transcendental equation (9), representations (7)–(11) enable a straightforward calculation of the GW amplitudes and group velocities in a prescribed observation direction u. III. COMPARISON WITH PLANE WAVE GROUP VELOCITY
In accordance with the modal analysis technique, a plane wave propagating over a laminate structure is sought in the form uðxÞ ¼ aðzÞeiðkx xþky yxtÞ :
(12)
The wave vector k ¼ ðkx ; ky Þ, kx ¼ k cos c, ky ¼ k sin c, k ¼ jkj, determines the orientation of the plane wave front, pointing out the direction of its movement in the plane ðx; yÞ. The substitution of Eq. (12) into the governing equations and homogeneous boundary conditions leads to the same characteristic equation (6) but relative to the wave vector components kx and ky instead of the Fourier parameters a1 and a2 . Its roots k ¼ kn ðc; xÞ or 150
x ¼ Wn ðk; cÞ
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
(13)
FIG. 2. (Color online) Geometric relation between (a) the wave vector curve and the group velocity vector cg , (b) the group velocity curve and the wave vector k; (c) out-of-plane CGW amplitudes scanned at the surface of a [0 ] composite plate illustrating central similarity of the group velocity and wave front curves. Glushkov et al.: Group velocity in anisotropic composites
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locus of cylindrical waves propagating from the origin with the group velocity cg ðcÞ [Fig. 2(c)], therefore, it is referred to as a wavefront curve as well.6 In the polar coordinates ðk; cÞ, @W 1 @W ; (14) cg ¼ grad W ¼ ¼ ðck ; cc Þ: @k k @c The components ck and cc are the cg projections onto the polar and angle axes ik and ic , respectively. Since ik is directed along the wave vector kðcÞ, the steering angle # may be calculated via the ratio of the catheti cc and ck : 1 @W @W : tg # ¼ cc =ck ¼ k @c @k The derivatives @W=@c and @W=@k are not independent. Differentiating x ¼ Wðk; cÞ with respect to c at a fixed x yields @W=@c ¼ ð@W=@kÞk0 ðcÞ. As a consequence, tg# ¼ k0 ðcÞ=kðcÞ; @W ð1; k0 =kÞ; @k @W qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 þ ðk0 =kÞ : cg ¼ jcg j ¼ @k
cg ¼
(15)
These expressions are consistent with the known, given in other notations, formulas.4–6 Let us show that the steering angle # and the magnitude cg of a PGW coincide with those of a CGW if the plane wave velocity vector cg points in the propagation direction u of the cylindrical wave [i.e., if w ¼ u, Fig. 2(c)]. In the plane wave notations fn ðhÞ ¼ kðcÞ, and phase equation (9) takes the form ctg bm ¼ k0 ðcm Þ=kðcm Þ;
cg pointing to the prescribed observation direction u. Therefore, one can use Eq. (9) in the context of modal analysis to select plane waves of form (12) carrying out wave envelopes in the direction required. Such an approach has been already used for the construction of 3D Green’s function of a composite plate in the form of superposition of two-dimensional (2D) plane waves with properly selected wave vectors k.10 The formulas derived in that work are in agreement with those presented in Eqs. (7)–(11). IV. NUMERICAL ILLUSTRATION AND EXPERIMENTAL VERIFICATION
In composite plates, the deviation of the GW group velocity cg from the wave vector k may be essential. Figure 3 demonstrates typical dependencies of the steering angle # on the k angle c. They were calculated using Eq. (15) for the fundamental A0 , S0 , and SH0 modes propagating in two composite plate samples with unidirectional lay-up [0 ] (left) and cross-ply lay-up ½0 ; 90 s (right). The mechanical properties of the samples’ pre-pregs are specified by the stiffness coefficients C11 ¼ 95:93, C12 ¼ 3:57, C22 ¼ C33 ¼ 9:61, C44 ¼ 3:0, C55 ¼ C66 ¼ 3:45 [GPa] and the density q ¼ 1482 kg/m3 ; the plate’s thickness is h ¼ 1:1 mm. The effective material properties were obtained based on the group velocity measurements and inverse problem solution, as described in Ref. 9. Solid lines in Fig. 3 are for #ðcÞ at the frequency f ¼ 50 kHz, the dashed ones are for f ¼ 200 kHz. One can see that the group velocity deviation of S0 and SH0 modes, which are practically frequency independent, may exceed 45 . The steering angle of A0 mode is not so large, but it changes with frequency, as previously noted in Ref. 5.
(16)
where cm ¼ hm ¼ bm þ u þ p=2. Since cm is the polar angle at which the wavenumber k is taken, the difference between the angles u and cm is, obviously, the steering angle #m for the mth cylindrical GW mode anm eisnm r related to the generating pole fn [see Eq. (7) and Fig. 2]. The substitution of bm ¼ cm u p=2 into Eq. (16) yields the same expression for #m ¼ u cm as Eq. (15) for the plane wave steering angle #: tg #m ¼ k0 ðcm Þ=kðcm Þ;
m ¼ 1; 2; :::; Mn :
(17)
In the same way, the group velocity magnitude of CGW, introduced in Eq. (11), can be represented in the form cg ðuÞ ¼ j@Wðk; cm Þ=@kjj1=cos #m j:
(18)
It is easily seen that jcos #m j1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1 þ tg2 #m ¼ 1 þ ðk0 =kÞ , which leads to the same value cg as in Eq. (15). A comparison of Eqs. (7) and (15)–(11) gives a physical meaning to the stationary points bm obtained from Eq. (9) or Eq. (16). They give the angles cm of the wave vectors kðcm Þ that specify plane waves (12) with the group velocity vectors J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
FIG. 3. (Color online) Steering angle # versus the wave vector’s polar angle c for the fundamental A0 , S0 , and SH0 GWs propagating over composite plates at the frequencies f ¼ 50 kHz (solid lines) and f ¼ 200 kHz (dashed lines). Glushkov et al.: Group velocity in anisotropic composites
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FIG. 4. (Color online) Group velocity polar plots cg ðwÞ for the fundamental A0 , S0 , and SH0 modes in [0 ] and [0 ,90 ]s plates; solid lines are for PGWs calculated using Eq. (15), circle markers are for CGWs, Eq. (11).
Figure 4 illustrates the coincidence of the PGW and CGW group velocity magnitudes calculated for these modes using Eq. (15) [cg ðwÞ, solid lines] and Eq. (11) [cg ðuÞ, circle markers], respectively. The derivative in Eq. (11) was calculated using a simple finite-difference approximation with a small frequency variation dx: cg
dx : snm ðx þ dxÞ snm ðxÞ
The group velocity curves for A0 and S0 modes (inner and outer convex curves in Fig. 4) are one-to-one dependencies on the observation angle u (Mn ¼ 1), while concavities of SH0 wave vector curves lead to well-known self-crossing loop patterns of the SH0 group velocity curves located between the A0 and S0 ones. It results in multivalued dependencies of SH0 group velocity on u (Mn ¼ 3) in the most part of the angle range. Nevertheless, the PGW and CGW group velocity curves cg ðwÞ and cg ðuÞ are identical for any mode, as it follows from the comparison with the plane wave formulas in Sec. III above. Experimental measurements were performed on the [0 ] composite plate specimen in order to test the practical ability of the CGW expressions to predict group velocity magnitudes in a prescribed observation direction u. GWs were excited in the plate by a small circular piezoceramic wafer (3-mm electrode radius, 0.25-mm thick, PIC151 ceramic; PI Ceramic GmbH, Lederhose, Germany) that was adhesively bonded on the top surface in the center of the plate (Fig. 1). A Tektronix AFG 3022B two-channel arbitrary signal generator (Tektronix Inc., Beaverton, OR) and a NF HSA 4014 amplifier (NF Corp., Yokohama, Japan) were used to generate 75 Vpp excitation to the piezoceramic wafer actuator. The excitation signal was a narrow square pulse (approximately 106 s) which provides a broadband signal covering a frequency range up to 1 MHz. The propagating waves were measured with a Polytec PSV-400 scanning laser (Polytec GmbH, Waldbronn, Germany) coupled with a Tektronix TDS 2014B four channel digital storage oscilloscope. The scanning head of the PSV-400 system was placed 1.17 m above the specimen. A thin reflective film was glued to the surface of the plate in the area of observation in order to improve the reflection of the laser beam and to maximize the signal-to-noise ratio. It was shown that the presence of such a tape did not 152
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
introduce unwanted effects into the propagating GW field.11 Our measurements, carried out with and without the film, also confirmed this fact. The experimental dependence of CGW wavenumbers snm on the frequency f for a fixed angle u was obtained using the spatial and temporal Fourier transforms. For this purpose, narrow long strips (typically 3 120 mm2 ) were scanned on the plate in the directions u ¼ 0 , 30 , 60 , and 90 . [Such a scanned area is marked schematically with red dots in Fig. 1(b).] The scanning area starts about 15 mm away from the actuator such that the point source approximation holds. The vibrometer head is placed above the middle of the area to minimize the discrepancy in S0 measurements.12 The 2D time-spatial Fourier transform ð1ð1 vz ðr; tÞeiðkrþ2pftÞ drdt (19) Hðk; f Þ ¼ 0
0
is applied to the experimentally acquired out-of-plane surface velocity vz . The polar radius r goes here along the central line of the scanned area as shown in Fig. 1(b). Local maxima of Hðk; f Þ give wavenumber dispersion curves in the wavenumber-frequency domain ðk; f Þ. The experimental results obtained for the unidirectional plate [0 ] in the selected directions u are shown in Fig. 5 as contour plots, where darker zones correspond to higher values of Hðk; f Þ. Theoretical curves of the CGW wavenumbers snm , given by Eq. (8) are overlapped in Fig. 5 as solid lines. One can see a good agreement between theoretical predictions and experimental results for both A0 and S0 modes. Regarding the SH0 modes, one notes that axially symmetric circular piezoactuators cannot excite shear horizontally polarized waves in isotropic plates. However, in the composite plates under consideration, the SH0 modes may appear due to the anisotropy. Their amplitudes are weak; in Fig. 5, they are visible only in the subplots for u ¼ 0 and 60 . The directions u ¼ 0 and u ¼ 90 coincide with the principal axes of anisotropy. In these directions the group velocity vector cg does not deviate from the wave vector k (# ¼ 0), and the dispersion dependence of the wave numbers snm for cylindrical waves and kn for plane waves is the same. In other directions they are different; to show the rate of this difference, the PGW dispersion curves kn ðf Þ are also put in Fig. 5 Glushkov et al.: Group velocity in anisotropic composites
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FIG. 5. (Color online) Level lines of function Hðk; f Þ calculated based on the experimental data acquired for various observation directions u. Solid lines are theoretical CGW dispersion curves; dashed lines are the curves kn ðf Þ for the corresponding PGW.
as dashed lines. Their deviation from the dark zones is appreciable, which confirms the theoretical conclusion that plane wave dispersion dependencies kn ðf Þ should not be used in anisotropic structures for the prediction of GW propagation characteristics in the direction pointed by the wave vector k. Another experimental procedure was adopted13 to measure group velocity dispersion characteristics for the selected observation angles. The out-of-plane displacement velocities were measured at two points A and B along every radial trajectory u ¼ const [Fig. 1(b)]. After that, a continuous wavelet transform with the Gabor mother wavelet was applied to these data. This wavelet is particularly attractive because it provides the best balance between the time and frequency resolutions using the smallest possible Heisenberg uncertainty box. The time of arrival of the wavelet-transform scalogram peaks yields the TOF from the source to the points A
and B at various central frequencies fc . It allows one to estið1Þ ð2Þ mate the values of group velocities cg and cg of wave packets registered at these points, as well as to add the third ð3Þ estimation cg ¼ jrB rA j=jtB tA j. The experimental group velocities are then evaluated as the arithmetical mean ð1Þ ð2Þ ð3Þ cg ðcg þ cg þ cg Þ=3. Figure 6 shows the dispersion curves of CGW group velocities for different directions of A0 and S0 wave propagation in the [0 ] sample. The theoretical curves calculated using Eq. (11) are shown as solid lines and the experimental values are put by circle markers. These velocities vary in a rather wide range from 2.6 to 8 km/s and from 0.5 to 1.5 km/s for the S0 and A0 modes, respectively. However, the discrepancy between the experimental and theoretical values generally do not exceed 5% for all directions and frequencies considered.
FIG. 6. (Color online) Group velocity dispersion curves calculated for selected directions u based on Eq. (11) (solid lines) and experimentally measured (circle markers); [0 ] composite plate.
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
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V. CONCLUSION
Theoretical and experimental results presented in the paper confirm that the explicit expression for GW group velocity derived from the far-field asymptotics for CGWs in anisotropic plates is valid for the group velocity magnitude and TOF evaluation of actual composites. Unlike plane wave representations, the cylindrical wave representation directly gives frequency dependent values (dispersion curves) of group velocities for wave packets propagating along a radial trajectory in any prescribed direction. ACKNOWLEDGMENTS
The work is partly supported by the Russian Foundation for Basic Research (RFBR, Project Nos. 12-01-33011 and 13-01-96516) and the German-Russian Interdisciplinary Science Center (G-RISC, Project No. M-2013a-4). The authors are grateful to T. Fedorkova and S. F€oll (HelmutSchmidt-University, Hamburg) for their help in conducting laser vibrometry experiments.
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Glushkov et al.: Group velocity in anisotropic composites
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Rolf Lammering Institute of Mechanics, Helmut-Schmidt-University/University of the Federal Armed Forces, Hamburg, D-22043, Germany
(Received 25 June 2013; revised 21 October 2013; accepted 28 October 2013) An explicit expression for the group velocity of wave packets, propagating in a laminate anisotropic composite plate in prescribed directions, is proposed. It is based on the cylindrical guided wave asymptotics derived from the path integral representation for wave fields generated in the composites by given localized sources. The expression derived is theoretically confirmed by the comparison with a known representation for the group velocity vector of a plane guided wave. Then it is experimentally validated against laser vibrometer measurements of guided wave packets generated by a piezoelectric wafer active sensor in a composite plate. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4829534] V PACS number(s): 43.35.Cg, 43.35.Pt, 43.20.Rz, 43.35.Zc [JBL]
I. INTRODUCTION
The study of elastic guided wave (GW) propagation in layered structures is important for such applications as geophysics, seismology, ultrasonic non-destructive testing, structural health monitoring (SHM), and design of piezoelectrically based microelectronic devices on surface acoustic waves. In SHM, damage detection in laminate plate-like structures is based on GW generation and sensing by piezoelectric wafer active sensors (PWASs) attached to the inspected structure. The GWs, scattered by a local structural inhomogeneity, reveal the presence of a flaw, while its location is estimated from the time of signal arrival (the time of flight—TOF) registered by the elements of a sensor network. The size and form of the flaw may be assessed through its imaging using various techniques, e.g., by the embedded ultrasonic structural radar.1 Most of the imaging algorithms rely on the TOF of transmitted and scattered signals, which depend on the distance and GW group velocities in the frequency band used.2 With isotropic structures those algorithms work well, since the wavenumbers, velocities, and dispersion dependencies for any direction of wave propagation are the same. With anisotropic composite plates, the algorithms of flaw detection are also based on the TOF estimation for plane guided waves (PGWs). However, here their characteristics are directionally dependent. In general, the directivity of the group velocity vector cg of a plane wave does not coincide with the directivity of the wave vector k,3 and the direction of wave packet propagation, specified by the vector cg , is not orthogonal to the wavefront associated with the wave vector k. Hence, for a GW packet propagating in the observation direction required, one has to tune the plane
a)
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Pages: 148–154
wave orientation before the group velocity calculation. The PGW directivity is frequency dependent4–6 that creates additional complications for the TOF evaluation and makes computer modeling much more time consuming. On the other hand, there is no necessity to use plane waves for the TOF evaluation in SHM, since the waves generated by comparatively small PWAS transducers or scattered by localized obstacles are cylindrical guided waves (CGWs). Moreover, the asymptotic expressions derived for CGWs generated by a given force source7 allow one to avoid those difficulties, as they provide a straightforward algorithm for the group velocity calculation in any prescribed direction. These asymptotics were numerically tested and experimentally validated in the course of investigation of frequency dependent CGW directivity8 and reconstruction of effective elastic moduli9 in composite plates. The present paper is focused on a theoretical and experimental validation of the CGW group velocity calculation based on these asymptotic representations. First, we repeat a summary of these expressions, while the exhaustive formulas and algorithms are available in Ref. 7. Then, we show that these formulas yield the same steering angles and group velocity magnitudes as the ones obtained for plane waves. Finally, wave number and group velocity dispersion curves for the fundamental A0 and S0 modes propagating from the source in various fixed directions are tested against experimental data obtained based on laser vibrometer scanning of PWAS generated out-of-plane GW amplitudes. II. CYLINDRICAL GUIDED WAVES
GW propagation caused in a laminate composite plate by a localized force source is described by the solution of a three-dimensional (3D) boundary value problem for the elastodynamic equations in displacements Cijkl ul;jk þ qx2 ui ¼ 0;
0001-4966/2014/135(1)/148/7/$30.00
i ¼ 1; 2; 3;
(1)
C 2014 Acoustical Society of America V
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given in each sublayer Dm of a multilayer domain D ¼ [M m¼1 Dm : jxj < 1; jyj < 1; H z 0 (Fig. 1) with specific values of the elastic stiffness tensor components Cijkl and material density q. The displacement vector u ¼ fu1 ; u2 ; u3 g is the complex amplitude of the steady-state harmonic oscillation ueixt with the angular frequency x ¼ 2pf ; f is frequency. Hereinafter, we try to keep the previous notations.7 The sublayers Dm are perfectly bonded with each other at the interfaces z ¼ zm , m ¼ 2; 3; :::; M, while the outer sides z ¼ z1 ¼ 0 and z ¼ zMþ1 ¼ H are stress-free except in an area X, where a surface load qðx; yÞeixt , simulating a source action (e.g., PWAS), is given. The response of the structure on the applied load is derived in terms of the convolution of its Green’s matrix kðx; xÞ with the vector-function q: ðð kðx n; y g; zÞqðn; gÞdndg: (2) uðx; xÞ ¼ X
The columns kj of the 3 3 Green’s matrix k ¼ ðk1 ⯗k2 ⯗k3 Þ are the displacement vectors corresponding to concentrated point loads applied at the origin O along the coordinate vectors ij , j ¼ 1; 2; 3. Alternatively, Eq. (2) may be written as
an inverse Fourier transform of the product of the Fourier symbols K ¼ F xy ½k and Q ¼ F xy ½q: uðx; xÞ ¼ F 1 xy ½KQ ð ð 1 Kða1 ; a2 ; zÞQða1 ; a2 Þ ¼ ð2pÞ2 C1 C2 eiða1 xþa2 yÞ da1 da2 :
(3)
The integration paths C1 and C2 go along the real axes deviating from them into the complex planes a1 and a2 for rounding real poles fn of the matrix K elements. We intentionally use the notations k and K for the Green’s matrices of layered structures and their Fourier symbols, to differ them from the classical matrices of fundamental solutions in the whole space. It coincides with the traditional wavenumber notation k ¼ jkj, but it should not be confusing for they appear in different contexts. If the space variables x ¼ ðx1 ; x2 ; x3 Þ ¼ ðx; y; zÞ and the Fourier parameters a ¼ ða1 ; a2 Þ are taken in the cylindrical and polar coordinates ðr; u; zÞ and ða; cÞ [see Fig. 1(b)]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ r cos u 2 þ y2 r ¼ pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ a cos c 2 2 ; y ¼ r sin u; a2 ¼ a sin c a ¼ a1 þ a2 0 c; u 2p; z¼z
(4)
then integral representation (3) takes the form ð ð 2p 1 Kða; c; zÞQða; cÞeiarcosðcuÞ dcada; uðxÞ ¼ ð2pÞ2 Cþ 0 (5) where Cþ is an integration contour, going in the complex plane a along the real semi-axis Re a 0, Im a ¼ 0 also bypassing real poles fn . These poles are roots of the characteristic equation Dða1 ; a2 ; xÞ ¼ Dða; c; xÞ ¼ 0;
(6)
where D is the denominator of the matrix K elements. To a constant factor, it coincides with the determinant of the matrix A resulting from the substitution of general solutions of Eq. (1) in the Fourier transform domain into the transformed boundary conditions at the surfaces and interfaces z ¼ zm , m ¼ 1; 2; ::::; M þ 1 [see Eqs. (38) and (39) in Ref. 7]. The application of the residual theorem and the stationary phase method to the integrals over a and c, respectively, leads to a far-field expansion in terms of CGWs. This expansion may be obtained in various forms. More precise formulas follow from the representations in terms of cylindrical Hankel functions Hmð1Þ ðfn rÞ [e.g., from Eq. (20) in Ref. 7], while a more physically evident one is uðxÞ ¼
N X un ðxÞ þ OððfrÞ1 Þ; fr ! 1; n¼1
FIG. 1. (Color online) Sketch of the problem: (a) Side view and (b) top view and coordinate systems. J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
un ðxÞ ¼
Mn X pffiffiffiffiffi anm ðu; zÞeisnm ðuÞr = fr:
(7)
m¼1
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Here N is the number of real poles fn ðc; xÞ, and f is a characteristic wavenumber. Every term un is associated with one generating pole fn , while, generally, it may consist of Mn > 1 terms describing CGWs with the amplitude factors anm and the wavenumbers snm : snm ðuÞ ¼ sn ðbnm Þ; sn ðbÞ ¼ fn ðhÞsinb; h ¼ b þ u þ p=2:
(8)
The number Mn of wavenumbers is determined by the number of roots bnm of the stationary phase equation fn ðhÞ cos b þ f0n ðhÞ sin b ¼ 0;
(9)
where f0n is the derivative of fn with respect to its angular argument. In the special case of isotropic materials, fn are angular independent, hence, f0n ¼ 0, Mn ¼ 1, bn1 ¼ p=2, and sn1 ¼ fn . The amplitude factors anm are expressed via the residues of K elements from the poles fn and the values of the Fourier symbol Qða; cÞ at the points a ¼ snm and c ¼ u: anm ¼ bnm ðu; zÞQðsnm ; uÞ; bnm ¼ jn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if=ð2ps00n ðbnm ÞÞ fn res Kða; hnm ; zÞja¼fn : (10)
The coefficient jn is nearly always jn ¼ 1, except in the case of infrequent irregular real poles yielding backward modes for which jn ¼ 1. Since the angular dependent values snm ðuÞ play the role of CGW wavenumbers, the group velocities of wave envelopes propagating in the u direction should be cg ðuÞ ¼ ½dsnm =dx1 :
are spectral points of the homogeneous problem under consideration. For any fixed angle c, they link k and x, specifying dispersion properties of the corresponding plane wave modes. Non-zero eigensolutions an ðzÞ, associated with these roots, are cross-sectional eigenforms of the related PGWs. Since the characteristic equation remains the same, the wavenumbers kn ðc; xÞ coincide with the poles fn ðc; xÞ in the CGW asymptotics above. It is worth noting again that the wavenumbers of the latter are not fn but snm of form (8). Let x ¼ WðkÞ ¼ Wðk; cÞ be one of the roots in the frequency domain. (To simplify notations, the subscript n is usually omitted.) The slowness, phase, and group velocity vectors of the related PGW are defined via W and k as4–6 s ¼ k=W;
cp ¼ ðW=k2 Þk
and
cg ¼ grad WðkÞ:
The vectors s and cp are collinear with k, while the group velocity cg may deviate from k. Being the gradient of WðkÞ, the vector cg is orthogonal to the wave-vector (or slowness) curve jkðcÞj, which is drawn in the wavenumber plane ðkx ; ky Þ by the tip of vector kðcÞ as c varies in the range 0 c 2p [Fig. 2(a)]. With an anisotropic sample, this curve ceases to be a simple circle, so that the polar angle wðcÞ of the vector cg ¼ cg ðcosw; sinwÞ, cg ¼ jcg j, generally differs from c. The angle # ¼ wðcÞ c of the deviation between cg and k is referred to as a steering angle.4–6 Reciprocally, the vector kðcÞ is orthogonal to the closed curve formed by the tip of vector cg ðwÞ, 0 w 2p [a group velocity curve, Fig. 2(b)]. Physically, it is centrally similar to the wavefront
(11)
Note that after the roots bnm are obtained from transcendental equation (9), representations (7)–(11) enable a straightforward calculation of the GW amplitudes and group velocities in a prescribed observation direction u. III. COMPARISON WITH PLANE WAVE GROUP VELOCITY
In accordance with the modal analysis technique, a plane wave propagating over a laminate structure is sought in the form uðxÞ ¼ aðzÞeiðkx xþky yxtÞ :
(12)
The wave vector k ¼ ðkx ; ky Þ, kx ¼ k cos c, ky ¼ k sin c, k ¼ jkj, determines the orientation of the plane wave front, pointing out the direction of its movement in the plane ðx; yÞ. The substitution of Eq. (12) into the governing equations and homogeneous boundary conditions leads to the same characteristic equation (6) but relative to the wave vector components kx and ky instead of the Fourier parameters a1 and a2 . Its roots k ¼ kn ðc; xÞ or 150
x ¼ Wn ðk; cÞ
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
(13)
FIG. 2. (Color online) Geometric relation between (a) the wave vector curve and the group velocity vector cg , (b) the group velocity curve and the wave vector k; (c) out-of-plane CGW amplitudes scanned at the surface of a [0 ] composite plate illustrating central similarity of the group velocity and wave front curves. Glushkov et al.: Group velocity in anisotropic composites
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locus of cylindrical waves propagating from the origin with the group velocity cg ðcÞ [Fig. 2(c)], therefore, it is referred to as a wavefront curve as well.6 In the polar coordinates ðk; cÞ, @W 1 @W ; (14) cg ¼ grad W ¼ ¼ ðck ; cc Þ: @k k @c The components ck and cc are the cg projections onto the polar and angle axes ik and ic , respectively. Since ik is directed along the wave vector kðcÞ, the steering angle # may be calculated via the ratio of the catheti cc and ck : 1 @W @W : tg # ¼ cc =ck ¼ k @c @k The derivatives @W=@c and @W=@k are not independent. Differentiating x ¼ Wðk; cÞ with respect to c at a fixed x yields @W=@c ¼ ð@W=@kÞk0 ðcÞ. As a consequence, tg# ¼ k0 ðcÞ=kðcÞ; @W ð1; k0 =kÞ; @k @W qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 þ ðk0 =kÞ : cg ¼ jcg j ¼ @k
cg ¼
(15)
These expressions are consistent with the known, given in other notations, formulas.4–6 Let us show that the steering angle # and the magnitude cg of a PGW coincide with those of a CGW if the plane wave velocity vector cg points in the propagation direction u of the cylindrical wave [i.e., if w ¼ u, Fig. 2(c)]. In the plane wave notations fn ðhÞ ¼ kðcÞ, and phase equation (9) takes the form ctg bm ¼ k0 ðcm Þ=kðcm Þ;
cg pointing to the prescribed observation direction u. Therefore, one can use Eq. (9) in the context of modal analysis to select plane waves of form (12) carrying out wave envelopes in the direction required. Such an approach has been already used for the construction of 3D Green’s function of a composite plate in the form of superposition of two-dimensional (2D) plane waves with properly selected wave vectors k.10 The formulas derived in that work are in agreement with those presented in Eqs. (7)–(11). IV. NUMERICAL ILLUSTRATION AND EXPERIMENTAL VERIFICATION
In composite plates, the deviation of the GW group velocity cg from the wave vector k may be essential. Figure 3 demonstrates typical dependencies of the steering angle # on the k angle c. They were calculated using Eq. (15) for the fundamental A0 , S0 , and SH0 modes propagating in two composite plate samples with unidirectional lay-up [0 ] (left) and cross-ply lay-up ½0 ; 90 s (right). The mechanical properties of the samples’ pre-pregs are specified by the stiffness coefficients C11 ¼ 95:93, C12 ¼ 3:57, C22 ¼ C33 ¼ 9:61, C44 ¼ 3:0, C55 ¼ C66 ¼ 3:45 [GPa] and the density q ¼ 1482 kg/m3 ; the plate’s thickness is h ¼ 1:1 mm. The effective material properties were obtained based on the group velocity measurements and inverse problem solution, as described in Ref. 9. Solid lines in Fig. 3 are for #ðcÞ at the frequency f ¼ 50 kHz, the dashed ones are for f ¼ 200 kHz. One can see that the group velocity deviation of S0 and SH0 modes, which are practically frequency independent, may exceed 45 . The steering angle of A0 mode is not so large, but it changes with frequency, as previously noted in Ref. 5.
(16)
where cm ¼ hm ¼ bm þ u þ p=2. Since cm is the polar angle at which the wavenumber k is taken, the difference between the angles u and cm is, obviously, the steering angle #m for the mth cylindrical GW mode anm eisnm r related to the generating pole fn [see Eq. (7) and Fig. 2]. The substitution of bm ¼ cm u p=2 into Eq. (16) yields the same expression for #m ¼ u cm as Eq. (15) for the plane wave steering angle #: tg #m ¼ k0 ðcm Þ=kðcm Þ;
m ¼ 1; 2; :::; Mn :
(17)
In the same way, the group velocity magnitude of CGW, introduced in Eq. (11), can be represented in the form cg ðuÞ ¼ j@Wðk; cm Þ=@kjj1=cos #m j:
(18)
It is easily seen that jcos #m j1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1 þ tg2 #m ¼ 1 þ ðk0 =kÞ , which leads to the same value cg as in Eq. (15). A comparison of Eqs. (7) and (15)–(11) gives a physical meaning to the stationary points bm obtained from Eq. (9) or Eq. (16). They give the angles cm of the wave vectors kðcm Þ that specify plane waves (12) with the group velocity vectors J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
FIG. 3. (Color online) Steering angle # versus the wave vector’s polar angle c for the fundamental A0 , S0 , and SH0 GWs propagating over composite plates at the frequencies f ¼ 50 kHz (solid lines) and f ¼ 200 kHz (dashed lines). Glushkov et al.: Group velocity in anisotropic composites
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FIG. 4. (Color online) Group velocity polar plots cg ðwÞ for the fundamental A0 , S0 , and SH0 modes in [0 ] and [0 ,90 ]s plates; solid lines are for PGWs calculated using Eq. (15), circle markers are for CGWs, Eq. (11).
Figure 4 illustrates the coincidence of the PGW and CGW group velocity magnitudes calculated for these modes using Eq. (15) [cg ðwÞ, solid lines] and Eq. (11) [cg ðuÞ, circle markers], respectively. The derivative in Eq. (11) was calculated using a simple finite-difference approximation with a small frequency variation dx: cg
dx : snm ðx þ dxÞ snm ðxÞ
The group velocity curves for A0 and S0 modes (inner and outer convex curves in Fig. 4) are one-to-one dependencies on the observation angle u (Mn ¼ 1), while concavities of SH0 wave vector curves lead to well-known self-crossing loop patterns of the SH0 group velocity curves located between the A0 and S0 ones. It results in multivalued dependencies of SH0 group velocity on u (Mn ¼ 3) in the most part of the angle range. Nevertheless, the PGW and CGW group velocity curves cg ðwÞ and cg ðuÞ are identical for any mode, as it follows from the comparison with the plane wave formulas in Sec. III above. Experimental measurements were performed on the [0 ] composite plate specimen in order to test the practical ability of the CGW expressions to predict group velocity magnitudes in a prescribed observation direction u. GWs were excited in the plate by a small circular piezoceramic wafer (3-mm electrode radius, 0.25-mm thick, PIC151 ceramic; PI Ceramic GmbH, Lederhose, Germany) that was adhesively bonded on the top surface in the center of the plate (Fig. 1). A Tektronix AFG 3022B two-channel arbitrary signal generator (Tektronix Inc., Beaverton, OR) and a NF HSA 4014 amplifier (NF Corp., Yokohama, Japan) were used to generate 75 Vpp excitation to the piezoceramic wafer actuator. The excitation signal was a narrow square pulse (approximately 106 s) which provides a broadband signal covering a frequency range up to 1 MHz. The propagating waves were measured with a Polytec PSV-400 scanning laser (Polytec GmbH, Waldbronn, Germany) coupled with a Tektronix TDS 2014B four channel digital storage oscilloscope. The scanning head of the PSV-400 system was placed 1.17 m above the specimen. A thin reflective film was glued to the surface of the plate in the area of observation in order to improve the reflection of the laser beam and to maximize the signal-to-noise ratio. It was shown that the presence of such a tape did not 152
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
introduce unwanted effects into the propagating GW field.11 Our measurements, carried out with and without the film, also confirmed this fact. The experimental dependence of CGW wavenumbers snm on the frequency f for a fixed angle u was obtained using the spatial and temporal Fourier transforms. For this purpose, narrow long strips (typically 3 120 mm2 ) were scanned on the plate in the directions u ¼ 0 , 30 , 60 , and 90 . [Such a scanned area is marked schematically with red dots in Fig. 1(b).] The scanning area starts about 15 mm away from the actuator such that the point source approximation holds. The vibrometer head is placed above the middle of the area to minimize the discrepancy in S0 measurements.12 The 2D time-spatial Fourier transform ð1ð1 vz ðr; tÞeiðkrþ2pftÞ drdt (19) Hðk; f Þ ¼ 0
0
is applied to the experimentally acquired out-of-plane surface velocity vz . The polar radius r goes here along the central line of the scanned area as shown in Fig. 1(b). Local maxima of Hðk; f Þ give wavenumber dispersion curves in the wavenumber-frequency domain ðk; f Þ. The experimental results obtained for the unidirectional plate [0 ] in the selected directions u are shown in Fig. 5 as contour plots, where darker zones correspond to higher values of Hðk; f Þ. Theoretical curves of the CGW wavenumbers snm , given by Eq. (8) are overlapped in Fig. 5 as solid lines. One can see a good agreement between theoretical predictions and experimental results for both A0 and S0 modes. Regarding the SH0 modes, one notes that axially symmetric circular piezoactuators cannot excite shear horizontally polarized waves in isotropic plates. However, in the composite plates under consideration, the SH0 modes may appear due to the anisotropy. Their amplitudes are weak; in Fig. 5, they are visible only in the subplots for u ¼ 0 and 60 . The directions u ¼ 0 and u ¼ 90 coincide with the principal axes of anisotropy. In these directions the group velocity vector cg does not deviate from the wave vector k (# ¼ 0), and the dispersion dependence of the wave numbers snm for cylindrical waves and kn for plane waves is the same. In other directions they are different; to show the rate of this difference, the PGW dispersion curves kn ðf Þ are also put in Fig. 5 Glushkov et al.: Group velocity in anisotropic composites
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FIG. 5. (Color online) Level lines of function Hðk; f Þ calculated based on the experimental data acquired for various observation directions u. Solid lines are theoretical CGW dispersion curves; dashed lines are the curves kn ðf Þ for the corresponding PGW.
as dashed lines. Their deviation from the dark zones is appreciable, which confirms the theoretical conclusion that plane wave dispersion dependencies kn ðf Þ should not be used in anisotropic structures for the prediction of GW propagation characteristics in the direction pointed by the wave vector k. Another experimental procedure was adopted13 to measure group velocity dispersion characteristics for the selected observation angles. The out-of-plane displacement velocities were measured at two points A and B along every radial trajectory u ¼ const [Fig. 1(b)]. After that, a continuous wavelet transform with the Gabor mother wavelet was applied to these data. This wavelet is particularly attractive because it provides the best balance between the time and frequency resolutions using the smallest possible Heisenberg uncertainty box. The time of arrival of the wavelet-transform scalogram peaks yields the TOF from the source to the points A
and B at various central frequencies fc . It allows one to estið1Þ ð2Þ mate the values of group velocities cg and cg of wave packets registered at these points, as well as to add the third ð3Þ estimation cg ¼ jrB rA j=jtB tA j. The experimental group velocities are then evaluated as the arithmetical mean ð1Þ ð2Þ ð3Þ cg ðcg þ cg þ cg Þ=3. Figure 6 shows the dispersion curves of CGW group velocities for different directions of A0 and S0 wave propagation in the [0 ] sample. The theoretical curves calculated using Eq. (11) are shown as solid lines and the experimental values are put by circle markers. These velocities vary in a rather wide range from 2.6 to 8 km/s and from 0.5 to 1.5 km/s for the S0 and A0 modes, respectively. However, the discrepancy between the experimental and theoretical values generally do not exceed 5% for all directions and frequencies considered.
FIG. 6. (Color online) Group velocity dispersion curves calculated for selected directions u based on Eq. (11) (solid lines) and experimentally measured (circle markers); [0 ] composite plate.
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
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V. CONCLUSION
Theoretical and experimental results presented in the paper confirm that the explicit expression for GW group velocity derived from the far-field asymptotics for CGWs in anisotropic plates is valid for the group velocity magnitude and TOF evaluation of actual composites. Unlike plane wave representations, the cylindrical wave representation directly gives frequency dependent values (dispersion curves) of group velocities for wave packets propagating along a radial trajectory in any prescribed direction. ACKNOWLEDGMENTS
The work is partly supported by the Russian Foundation for Basic Research (RFBR, Project Nos. 12-01-33011 and 13-01-96516) and the German-Russian Interdisciplinary Science Center (G-RISC, Project No. M-2013a-4). The authors are grateful to T. Fedorkova and S. F€oll (HelmutSchmidt-University, Hamburg) for their help in conducting laser vibrometry experiments.
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Glushkov et al.: Group velocity in anisotropic composites
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