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The transient response of a transversely isotropic cylinder under a laser point source impact
Ultrasonics 44 (2006) e823–e827 www.elsevier.com/locate/ultras
The transient response of a transversely isotropic cylinder under a laser point source impact Y. Pan b
a,*
, M. Perton b, C. Rossignol b, B. Audoin
b
a Institute of Acoustics, Tongji University, 200092 Shanghai, PR China Laboratoire de Me´canique Physique, UMR CNRS no. 5469, Universite´ Bordeaux 1, 351 Cours de la Libe´ration, 33405 Talence, France
Available online 9 June 2006
Abstract The transient response of a transversely isotropic cylinder under a laser point source impact is solved theoretically. The radial displacement generated by the laser under the ablation regime is numerically calculated by introducing Fourier series expansion and two-dimensional Fourier transform. The validity of this theoretical solution is demonstrated on a fiber reinforced composite cylinder with a strong anisotropy. Experimental displacements are detected at the cylinder surface by the laser ultrasonic technique, and are analyzed by the ray trajectories. Corresponding theoretical displacements are calculated numerically and compared to the experimental signals. Good agreement is found. The diffraction effect caused by the cusp is observed in both theory and experiment. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Laser ultrasonics; Cylinder; Anisotropy
1. Introduction Many cylindrical components, such as fibers, wires, and rods are usually transversely isotropic (TI) due to their manufacturing processes. Transverse isotropy is often desirable for many structural applications, and thus it is important to study the wave propagation in such a cylinder. Early in 1965, Mirsky [1] theoretically investigated free harmonic vibrations of a cylinder. He found the frequency equation, i.e., the characteristic or dispersion equation for steady-state vibrations, and some dispersion curves were numerically calculated. A steady-state wave propagation was also documented by Payton [2]. For NDE purposes, the acoustic scattering of a TI cylinder has been extensively studied. Honarvar and Sinclair have tried to evaluate the elastic properties of a TI cylinder
*
Corresponding author. Fax: 86 21 6598 2314. E-mail address: [email protected] (Y. Pan).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.180
by experimentally observing the resonance spectrum resulted from the oblique insonification of a plane acoustic wave [3]. For this technique to be successfully employed, the sample must be immersed in a fluid, usually water. This requirement limits the capability of any ultrasonic immersion method when measurements need to be performed, for instance, at elevated temperatures. Fortunately, this difficulty can be overcome if laser-based ultrasonic techniques are used [4], in which the ultrasonic waves are generated and detected at a distance, without any contact to the sample. Recently, authors have published a two-dimensional (2D) model [5] to calculate the transient response of a TI cylinder impacted by a laser line pulse. Obviously, this model cannot predict the wave propagation along the cylinder axis. A three-dimensional (3D) model [6] was developed soon after to calculate the transient response of an isotropic cylinder impacted by a laser point pulse. Therefore, anisotropy was not accounted for in that work. To the best of author’s knowledge, no work on the 3D transient response of a TI cylinder has been reported in the literature yet.
e824
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
(a)
(b) 10
axis z (km/s)
8
L
6
4
Tq
2
Tp 0 0
15
α
1
2 3 4 axis r (km/s)
5
Fig. 1. The problem geometry (a) and group velocity curves (b) for a transversely isotropic cylinder.
In this paper, a theoretical solution is presented to calculate the 3D transient response of a TI cylinder impacted by a laser point pulse. Experimental responses are detected by the laser ultrasonic technique for a fiber reinforced composite cylinder. The theoretical responses are calculated and compared with the corresponding experimental ones. The effect of anisotropy is also discussed. 2. Theoretical solution A homogenous and TI cylinder of infinite length and radius a is considered. As shown in Fig. 1(a), the symmetry axis of the cylinder is assumed to coincide with the z-axis of the system of cylindrical coordinate (r, h, z). A pulsed laser beam focused by a spherical lens impacts the cylinder with a spot size of diameter 0.2 mm. The transient response is experimentally observed with a CW laser interferometer at the surface. The material considered in this work shows isotropy in (r, h) plane and anisotropy in (r, z) plane. Therefore, five independent elastic constants are required. In Voigt notation, they are c11 = 12.26 + 0.043i GPa, c13 = 5.93 + 0.016i GPa, c33 = 133.81 + 0.4i GPa, c44 = 6.0 + 0.015i GPa, c66 = 7.08 + 0.027i GPa, the imaginary parts accounting for the material’s viscoelasticity. The mass density of the fiber reinforced composite cylinder considered in this paper is 1.6 g/cm3. The group wave velocity curves for this material are shown in Fig. 1(b). The cuspidal shape (with one edge is near 15°) is usually observed for the vertically polarized quasi transverse wave (Tq). Other two curves represent the longitudinal (L) and the horizontally polarized pure transverse (Tq) waves respectively. Given the small optical penetration with respect to the acoustic wavelengths, the effect of the volume source can be neglected. The source is thus considered to lie on the cylinder surface and it can be taken account in the boundary conditions. Since a spherical lens is used to focus the
laser beam, the spatial acoustic source can be modeled as a point on the surface. Apart from this line, the cylinder surface is assumed to be stress-free. For the impact in ablation regime, vaporization of a small amount of surface material occurs due to high laser power intensity. The source is then modeled as a normal loading with a delta function of time. To derive the wave motion equations for the anisotropic cylinder, the same expression of the displacement in terms of three potentials [6,7] is considered to represent Tp wave, as well as the coupled L and Tq waves. Introducing the definition of the Fourier series expansion for coordinate h as in a recent work [8], and applying the 2D Fourier transform for time t and coordinate z, the wave motion equations and the boundary equations can be linearized, providing explicit solution forms for the potentials and thus for the displacement. For the considered ablation regime, the radial displacement on the surface is ur ða; h; tÞ ¼ ð2pÞ
3
Z
þ1 1
Z
þ1 1
ejðkzxtÞ dk dx;
þ1 X
! U r ða; m; xÞe
jmh
v¼1
ð1Þ
where U r ða; m; k; xÞ ¼
F 0a fA1 B1 jkaq2 A2 B2 jvA3 g; 2c66 Dðm; k; xÞ ð2Þ
Here, the wave number m = kha, the angular frequency x, and the wave component k are thus introduced. kh, k are components of the wave vector k along the h and z directions, respectively. F0 is a certain loading in N ls m1. In Eq. (2), the coefficients A1, A2, and A3 are determined by the elements in the following matrix:
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
2
e825
jka ðq2 c11 b22 a2 c13 Þ=2c66 q2 ðm2 B2 Þ vkaq2 ð1 B2 Þ
m2 B1 ðc11 b21 þ q1 c13 k 2 Þa2 =2c66 6 fmij g ¼ 4 jmð1 B1 Þ
2 2
jkað1 þ q1 ÞB1 c44 =2c66
3 jmð1 B3 Þ 7 c2 a2 =2 m2 þ B3 5:
ð1 k a q2 ÞB2 c44 =2c66
H Tq T Tq Tq p
L LL Tq T T q q
10
mkac44 =2c66
Tp Tp
R
8
z (mm)
6
4
2
D LL T L q
0 1
2
Tp
3
4
LL
q q
(a)
10
H Tp Tp
Tq Tq
5 time ( μ s)
7
6
q
Tq T T
L
R
Tp
8
9
Tp Tp
T Tq Tq
H
R
8
z (mm)
6
4
2
0
LD 1
(b)
2
LL Tq 3
Tp T T H q q 4
R
Tp Tp 5
6
7
8
9
time ( μ s)
Fig. 2. Waterfall plots of (a) experimental and (b) theoretical displacements at z = 0, 0.4, . . . , and 9.2 mm observation positions.
ð3Þ
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Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
and, in particular, A1 ¼ ðm22 m33 m23 m32 Þ; A2 ¼ ðm23 m31 m21 m33 Þ;
A3 ¼ ðm21 m32 m22 m31 Þ:
ð4Þ
Note that in Eq. (2), D(m, k, x) = det(mij) is the determinant of the matrix {mij}. Also the coefficients q1 and q2 are given by 8 ðc13 þ c44 Þðqx2 c44 k 2 Þ > > ; > q1 ¼ < ðc13 þ c44 Þ2 k 2 þ c11 ðqx2 c33 k 2 Þ b21 c11 c44 ð5Þ > ðc13 þ c44 Þ > > ; : q2 ¼ ðc11 b22 þ c44 k 2 qx2 Þa2
the evolution of the shape. Here dashed lines display the time arrivals calculated based on the corresponding ray trajectories (see Fig. 6 of Ref. [6]) of identified waves for an isotropic cylinder. Since the cylindrical Rayleigh wave is dispersive, its group wave velocity is approximated by that of an anisotropic halfspace. The direct longitudinal (L) and vertically polarized quasi transverse (Tq) waves are clearly observable. Their once-reflected counterparts (LL) and (TqTq) are observable with decreased amplitudes mainly due to the different directivities. Both Tq and TqTq waves show the similar
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 b1 ¼ ðE DÞ=ð2c11 c44 Þ; > > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 ¼ ðE þ DÞ=ð2c11 c44 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : c ¼ ðqx2 c44 k 2 Þ=c66
z = 0 mm
displacement (pm)
In Eq. (2), the parameters B1 and B2, and B3 are determined 8 0 > < B1 ¼ b1 aJ t ðb1 aÞ=J t ðb1 aÞ; B2 ¼ b2 aJ 0t ðb2 aÞ=J t ðb2 aÞ; ð6Þ > : 0 B3 ¼ caJ t ðcaÞ=J t ðcaÞ;
H R TpTp
L
10
LL
ð7Þ
R
L
TpTp
10
and 8 < E ¼ ðc13 þ c44 Þ2 k 2 þ c11 ðqx2 c33 k 2 Þ þ c44 ðqx2 c44 k 2 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : D ¼ E2 4c11 c44 ðqx2 c33 k 2 Þðqx2 c44 k 2 Þ: ð8Þ
H
LL
1
2
3
4
In Eqs. (6) and (7), J 0m ðxÞ is the derivative of the Bessel function Jv(x); b1, b2, and c are the wave numbers for the L, Tq, and Tp waves, respectively. A small constant d is introduced as the imaginary part of x as discussed in a recent paper [8]. Doing so, x becomes complex, x jd, and the poles of the determinant are moved off the real axis of variable m. Now the integration can be calculated numerically. The value d = 0.06 rad ls1 is chosen for the auxiliary parameter in the following numerical calculations.
5
6
7
8
time (μs)
(a)
z = 1.6 mm (α=15°)
displacement (pm) H
R
10
Tp T p
Tp
D Tq
T q Tq
L LL
R
3. Numerical and experimental results The laser ultrasonic system [4] was setup to observe the transient response experimentally. The system was first adjusted to locate the epicenter detection position, which corresponds to r = a, h = 180°, and z = 0 for the theoretical solution. The non-epicenter detection positions were realized by scanning the laser source by step of 0.4 mm along the z direction of the cylinder sample with the fixed angle h = 180°. Each signal was averaged by 20 shots. As shown in Fig. 2(a) and (b), the experimental and theoretical responses for any observation positions are in good agreement concerning time arrival, relative amplitude, and
H
D
10 L
Tp Tq T q Tq
Tp Tp
LL
1
(b)
2
3
4
5 time (μs)
6
7
8
Fig. 3. Experimental (up) and calculated (down) displacements at (a) the epicenter (a = 0°) and (b) a non-epicenter (a = 15°) positions.
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
cusps as in Fig. 1(b). These cusps disappear cylinder when the cylinder is isotropic [6]. Moreover, the direct horizontally polarized pure transverse wave (Tq) is less observable than the Tq wave, because its amplitude contributes less along the radial direction due to its polarization. Its oncereflected counterpart (TpTp) is more observable, as the polarization favors the radial detection after the reflection. A head wave (H) resulted from Tp wave is observed. The cylindrical Rayleigh wave (R) and its dispersive behavior are clearly discernible. Other waves such as the twicereflected longitudinal wave (3L) observed in an isotropic cylinder [6], or the head waves resulted from Tq wave are not illustrated here. Having low amplitude, they are possibly overlapped by other waves marked in Fig. 2(a) and (b). The experimental and theoretical responses are further compared at the epicenter (z = 0) and a non-epicenter (z = 1.6 mm) positions of the laser point detection, respectively. The theoretical waveforms were scaled vertically to account for the source strength. As shown in Fig. 3(a) and (b), they are in good agreement. The time, shape and relative amplitude of the marked wave arrivals are almost identical. The L, LL, TpTp, H, and R waves are clearly observable. At the epicenter, the theoretical response (solid line) obtained by the current 3D model is almost identical to that calculated with the previous 2D model [5] (dashed line). The slight deviation is due to the different directivity of the source. Moreover, the direct transverse waves Tp and Tq almost disappear, because the corresponding directivity has no contribution along this direction [4]. As the directivity shows non-zero amplitude, they appear for a nonepicenter observation. Here, the small arrival difference for R wave is caused by a tiny aluminum film stuck on the sample surface to improve the signal to noise ratio. Fig. 3(b) clearly reveals a wave denoted as D, which does not exist in the isotropic case [6]. This non-epicenter position corresponds to an observation angle of a = 15° shown as a dashed line in Fig. 1(b), which is close to the sharp point of the cuspidal shape. The velocity of this wave is identical to the value marked as a cross on the dashed line. This wave, resulting from the proximity of the cusp, cannot be predicted by the ray theory [9], since it does not originate from a homogeneous wave, but instead from an inhomogeneous one. This phenomenon is the so-called diffraction by the cusp edges, which does not exist for an isotropic material. The effect of anisotropy is clearly observed in both theory and experiment.
e827
4. Conclusion A theoretical solution is found to calculate the transient response of a transversely isotropic cylinder under a laser point source. Experimental and theoretical responses are obtained and compared for a fiber reinforced composite cylinder with a strong anisotropy. Good agreement is observed in arrival time, shape and relative amplitude (i) of the cylindrical Rayleigh waves, and (ii) of the various longitudinal and transverse bulk waves propagating through the cylinder or reflected at the free circular surface. The ansotropic effect is observed for both theory and experiment. The results will be useful to identify waves for solving an inverse problem, such as the elastic constants measurement. Acknowledgements This work was done at Laboratoire de Me´canique Physique, Universite´ Bordeaux 1. Y.P. was supported by CNRS K.C. Wong fellowships and by Natural Science Foundation of China under Grant No. 10234020. References [1] I. Mirsky, Wave propagation in transversely isotropic circular cylinders: theory and numerical results, J. Acoust. Soc. Am. 37 (1965) 1016– 1026. [2] R.G. Payton, Elastic Wave Propagation in Transversely Isotropic Media, Nijhoff, The Hague, The Netherlands, 1983. [3] F. Honarvar, A.N. Sinclair, Nondestructive evaluation of cylindrical components by resonance acoustic spectroscopy, Ultrasonics 36 (1998) 845–854. [4] C.B. Scruby, L.E. Drain, Laser Ultrasonics: Techniques and Applications, Adam Hilger, New York, 1990. [5] Y. Pan, C. Rossignol, B. Audoin, Acoustic waves generated by a laser line pulse in a transversely isotropic cylinder, Appl. Phys. Lett. 82 (2003) 4379–4381. [6] Y. Pan, C. Rossignol, B. Audoin, Acoustic waves generated by a laser point source in an isotropic cylinder, J. Acoust. Soc. Am. 116 (2004) 814–820. [7] A. Rahman, F. Ahmad, Representation of the displacement in terms of scalar functions for use in transversely isotropic materials, J. Acoust. Soc. Am. 104 (1998) 3675–3676. [8] Y. Pan, C. Rossignol, B. Audoin, Identification of laser generated acoustic waves in the 2D transient response of cylinders, J. Acoust. Soc. Am. 117 (6) (2005) 3600–3608. [9] H.J. Maris, Effect of finite phonon wavelength on phonon focusing, Phys. Rev. B 28 (12) (1983) 7033–7037.
The transient response of a transversely isotropic cylinder under a laser point source impact Y. Pan b
a,*
, M. Perton b, C. Rossignol b, B. Audoin
b
a Institute of Acoustics, Tongji University, 200092 Shanghai, PR China Laboratoire de Me´canique Physique, UMR CNRS no. 5469, Universite´ Bordeaux 1, 351 Cours de la Libe´ration, 33405 Talence, France
Available online 9 June 2006
Abstract The transient response of a transversely isotropic cylinder under a laser point source impact is solved theoretically. The radial displacement generated by the laser under the ablation regime is numerically calculated by introducing Fourier series expansion and two-dimensional Fourier transform. The validity of this theoretical solution is demonstrated on a fiber reinforced composite cylinder with a strong anisotropy. Experimental displacements are detected at the cylinder surface by the laser ultrasonic technique, and are analyzed by the ray trajectories. Corresponding theoretical displacements are calculated numerically and compared to the experimental signals. Good agreement is found. The diffraction effect caused by the cusp is observed in both theory and experiment. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Laser ultrasonics; Cylinder; Anisotropy
1. Introduction Many cylindrical components, such as fibers, wires, and rods are usually transversely isotropic (TI) due to their manufacturing processes. Transverse isotropy is often desirable for many structural applications, and thus it is important to study the wave propagation in such a cylinder. Early in 1965, Mirsky [1] theoretically investigated free harmonic vibrations of a cylinder. He found the frequency equation, i.e., the characteristic or dispersion equation for steady-state vibrations, and some dispersion curves were numerically calculated. A steady-state wave propagation was also documented by Payton [2]. For NDE purposes, the acoustic scattering of a TI cylinder has been extensively studied. Honarvar and Sinclair have tried to evaluate the elastic properties of a TI cylinder
*
Corresponding author. Fax: 86 21 6598 2314. E-mail address: [email protected] (Y. Pan).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.180
by experimentally observing the resonance spectrum resulted from the oblique insonification of a plane acoustic wave [3]. For this technique to be successfully employed, the sample must be immersed in a fluid, usually water. This requirement limits the capability of any ultrasonic immersion method when measurements need to be performed, for instance, at elevated temperatures. Fortunately, this difficulty can be overcome if laser-based ultrasonic techniques are used [4], in which the ultrasonic waves are generated and detected at a distance, without any contact to the sample. Recently, authors have published a two-dimensional (2D) model [5] to calculate the transient response of a TI cylinder impacted by a laser line pulse. Obviously, this model cannot predict the wave propagation along the cylinder axis. A three-dimensional (3D) model [6] was developed soon after to calculate the transient response of an isotropic cylinder impacted by a laser point pulse. Therefore, anisotropy was not accounted for in that work. To the best of author’s knowledge, no work on the 3D transient response of a TI cylinder has been reported in the literature yet.
e824
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
(a)
(b) 10
axis z (km/s)
8
L
6
4
Tq
2
Tp 0 0
15
α
1
2 3 4 axis r (km/s)
5
Fig. 1. The problem geometry (a) and group velocity curves (b) for a transversely isotropic cylinder.
In this paper, a theoretical solution is presented to calculate the 3D transient response of a TI cylinder impacted by a laser point pulse. Experimental responses are detected by the laser ultrasonic technique for a fiber reinforced composite cylinder. The theoretical responses are calculated and compared with the corresponding experimental ones. The effect of anisotropy is also discussed. 2. Theoretical solution A homogenous and TI cylinder of infinite length and radius a is considered. As shown in Fig. 1(a), the symmetry axis of the cylinder is assumed to coincide with the z-axis of the system of cylindrical coordinate (r, h, z). A pulsed laser beam focused by a spherical lens impacts the cylinder with a spot size of diameter 0.2 mm. The transient response is experimentally observed with a CW laser interferometer at the surface. The material considered in this work shows isotropy in (r, h) plane and anisotropy in (r, z) plane. Therefore, five independent elastic constants are required. In Voigt notation, they are c11 = 12.26 + 0.043i GPa, c13 = 5.93 + 0.016i GPa, c33 = 133.81 + 0.4i GPa, c44 = 6.0 + 0.015i GPa, c66 = 7.08 + 0.027i GPa, the imaginary parts accounting for the material’s viscoelasticity. The mass density of the fiber reinforced composite cylinder considered in this paper is 1.6 g/cm3. The group wave velocity curves for this material are shown in Fig. 1(b). The cuspidal shape (with one edge is near 15°) is usually observed for the vertically polarized quasi transverse wave (Tq). Other two curves represent the longitudinal (L) and the horizontally polarized pure transverse (Tq) waves respectively. Given the small optical penetration with respect to the acoustic wavelengths, the effect of the volume source can be neglected. The source is thus considered to lie on the cylinder surface and it can be taken account in the boundary conditions. Since a spherical lens is used to focus the
laser beam, the spatial acoustic source can be modeled as a point on the surface. Apart from this line, the cylinder surface is assumed to be stress-free. For the impact in ablation regime, vaporization of a small amount of surface material occurs due to high laser power intensity. The source is then modeled as a normal loading with a delta function of time. To derive the wave motion equations for the anisotropic cylinder, the same expression of the displacement in terms of three potentials [6,7] is considered to represent Tp wave, as well as the coupled L and Tq waves. Introducing the definition of the Fourier series expansion for coordinate h as in a recent work [8], and applying the 2D Fourier transform for time t and coordinate z, the wave motion equations and the boundary equations can be linearized, providing explicit solution forms for the potentials and thus for the displacement. For the considered ablation regime, the radial displacement on the surface is ur ða; h; tÞ ¼ ð2pÞ
3
Z
þ1 1
Z
þ1 1
ejðkzxtÞ dk dx;
þ1 X
! U r ða; m; xÞe
jmh
v¼1
ð1Þ
where U r ða; m; k; xÞ ¼
F 0a fA1 B1 jkaq2 A2 B2 jvA3 g; 2c66 Dðm; k; xÞ ð2Þ
Here, the wave number m = kha, the angular frequency x, and the wave component k are thus introduced. kh, k are components of the wave vector k along the h and z directions, respectively. F0 is a certain loading in N ls m1. In Eq. (2), the coefficients A1, A2, and A3 are determined by the elements in the following matrix:
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
2
e825
jka ðq2 c11 b22 a2 c13 Þ=2c66 q2 ðm2 B2 Þ vkaq2 ð1 B2 Þ
m2 B1 ðc11 b21 þ q1 c13 k 2 Þa2 =2c66 6 fmij g ¼ 4 jmð1 B1 Þ
2 2
jkað1 þ q1 ÞB1 c44 =2c66
3 jmð1 B3 Þ 7 c2 a2 =2 m2 þ B3 5:
ð1 k a q2 ÞB2 c44 =2c66
H Tq T Tq Tq p
L LL Tq T T q q
10
mkac44 =2c66
Tp Tp
R
8
z (mm)
6
4
2
D LL T L q
0 1
2
Tp
3
4
LL
q q
(a)
10
H Tp Tp
Tq Tq
5 time ( μ s)
7
6
q
Tq T T
L
R
Tp
8
9
Tp Tp
T Tq Tq
H
R
8
z (mm)
6
4
2
0
LD 1
(b)
2
LL Tq 3
Tp T T H q q 4
R
Tp Tp 5
6
7
8
9
time ( μ s)
Fig. 2. Waterfall plots of (a) experimental and (b) theoretical displacements at z = 0, 0.4, . . . , and 9.2 mm observation positions.
ð3Þ
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Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
and, in particular, A1 ¼ ðm22 m33 m23 m32 Þ; A2 ¼ ðm23 m31 m21 m33 Þ;
A3 ¼ ðm21 m32 m22 m31 Þ:
ð4Þ
Note that in Eq. (2), D(m, k, x) = det(mij) is the determinant of the matrix {mij}. Also the coefficients q1 and q2 are given by 8 ðc13 þ c44 Þðqx2 c44 k 2 Þ > > ; > q1 ¼ < ðc13 þ c44 Þ2 k 2 þ c11 ðqx2 c33 k 2 Þ b21 c11 c44 ð5Þ > ðc13 þ c44 Þ > > ; : q2 ¼ ðc11 b22 þ c44 k 2 qx2 Þa2
the evolution of the shape. Here dashed lines display the time arrivals calculated based on the corresponding ray trajectories (see Fig. 6 of Ref. [6]) of identified waves for an isotropic cylinder. Since the cylindrical Rayleigh wave is dispersive, its group wave velocity is approximated by that of an anisotropic halfspace. The direct longitudinal (L) and vertically polarized quasi transverse (Tq) waves are clearly observable. Their once-reflected counterparts (LL) and (TqTq) are observable with decreased amplitudes mainly due to the different directivities. Both Tq and TqTq waves show the similar
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 b1 ¼ ðE DÞ=ð2c11 c44 Þ; > > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 ¼ ðE þ DÞ=ð2c11 c44 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : c ¼ ðqx2 c44 k 2 Þ=c66
z = 0 mm
displacement (pm)
In Eq. (2), the parameters B1 and B2, and B3 are determined 8 0 > < B1 ¼ b1 aJ t ðb1 aÞ=J t ðb1 aÞ; B2 ¼ b2 aJ 0t ðb2 aÞ=J t ðb2 aÞ; ð6Þ > : 0 B3 ¼ caJ t ðcaÞ=J t ðcaÞ;
H R TpTp
L
10
LL
ð7Þ
R
L
TpTp
10
and 8 < E ¼ ðc13 þ c44 Þ2 k 2 þ c11 ðqx2 c33 k 2 Þ þ c44 ðqx2 c44 k 2 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : D ¼ E2 4c11 c44 ðqx2 c33 k 2 Þðqx2 c44 k 2 Þ: ð8Þ
H
LL
1
2
3
4
In Eqs. (6) and (7), J 0m ðxÞ is the derivative of the Bessel function Jv(x); b1, b2, and c are the wave numbers for the L, Tq, and Tp waves, respectively. A small constant d is introduced as the imaginary part of x as discussed in a recent paper [8]. Doing so, x becomes complex, x jd, and the poles of the determinant are moved off the real axis of variable m. Now the integration can be calculated numerically. The value d = 0.06 rad ls1 is chosen for the auxiliary parameter in the following numerical calculations.
5
6
7
8
time (μs)
(a)
z = 1.6 mm (α=15°)
displacement (pm) H
R
10
Tp T p
Tp
D Tq
T q Tq
L LL
R
3. Numerical and experimental results The laser ultrasonic system [4] was setup to observe the transient response experimentally. The system was first adjusted to locate the epicenter detection position, which corresponds to r = a, h = 180°, and z = 0 for the theoretical solution. The non-epicenter detection positions were realized by scanning the laser source by step of 0.4 mm along the z direction of the cylinder sample with the fixed angle h = 180°. Each signal was averaged by 20 shots. As shown in Fig. 2(a) and (b), the experimental and theoretical responses for any observation positions are in good agreement concerning time arrival, relative amplitude, and
H
D
10 L
Tp Tq T q Tq
Tp Tp
LL
1
(b)
2
3
4
5 time (μs)
6
7
8
Fig. 3. Experimental (up) and calculated (down) displacements at (a) the epicenter (a = 0°) and (b) a non-epicenter (a = 15°) positions.
Y. Pan et al. / Ultrasonics 44 (2006) e823–e827
cusps as in Fig. 1(b). These cusps disappear cylinder when the cylinder is isotropic [6]. Moreover, the direct horizontally polarized pure transverse wave (Tq) is less observable than the Tq wave, because its amplitude contributes less along the radial direction due to its polarization. Its oncereflected counterpart (TpTp) is more observable, as the polarization favors the radial detection after the reflection. A head wave (H) resulted from Tp wave is observed. The cylindrical Rayleigh wave (R) and its dispersive behavior are clearly discernible. Other waves such as the twicereflected longitudinal wave (3L) observed in an isotropic cylinder [6], or the head waves resulted from Tq wave are not illustrated here. Having low amplitude, they are possibly overlapped by other waves marked in Fig. 2(a) and (b). The experimental and theoretical responses are further compared at the epicenter (z = 0) and a non-epicenter (z = 1.6 mm) positions of the laser point detection, respectively. The theoretical waveforms were scaled vertically to account for the source strength. As shown in Fig. 3(a) and (b), they are in good agreement. The time, shape and relative amplitude of the marked wave arrivals are almost identical. The L, LL, TpTp, H, and R waves are clearly observable. At the epicenter, the theoretical response (solid line) obtained by the current 3D model is almost identical to that calculated with the previous 2D model [5] (dashed line). The slight deviation is due to the different directivity of the source. Moreover, the direct transverse waves Tp and Tq almost disappear, because the corresponding directivity has no contribution along this direction [4]. As the directivity shows non-zero amplitude, they appear for a nonepicenter observation. Here, the small arrival difference for R wave is caused by a tiny aluminum film stuck on the sample surface to improve the signal to noise ratio. Fig. 3(b) clearly reveals a wave denoted as D, which does not exist in the isotropic case [6]. This non-epicenter position corresponds to an observation angle of a = 15° shown as a dashed line in Fig. 1(b), which is close to the sharp point of the cuspidal shape. The velocity of this wave is identical to the value marked as a cross on the dashed line. This wave, resulting from the proximity of the cusp, cannot be predicted by the ray theory [9], since it does not originate from a homogeneous wave, but instead from an inhomogeneous one. This phenomenon is the so-called diffraction by the cusp edges, which does not exist for an isotropic material. The effect of anisotropy is clearly observed in both theory and experiment.
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4. Conclusion A theoretical solution is found to calculate the transient response of a transversely isotropic cylinder under a laser point source. Experimental and theoretical responses are obtained and compared for a fiber reinforced composite cylinder with a strong anisotropy. Good agreement is observed in arrival time, shape and relative amplitude (i) of the cylindrical Rayleigh waves, and (ii) of the various longitudinal and transverse bulk waves propagating through the cylinder or reflected at the free circular surface. The ansotropic effect is observed for both theory and experiment. The results will be useful to identify waves for solving an inverse problem, such as the elastic constants measurement. Acknowledgements This work was done at Laboratoire de Me´canique Physique, Universite´ Bordeaux 1. Y.P. was supported by CNRS K.C. Wong fellowships and by Natural Science Foundation of China under Grant No. 10234020. References [1] I. Mirsky, Wave propagation in transversely isotropic circular cylinders: theory and numerical results, J. Acoust. Soc. Am. 37 (1965) 1016– 1026. [2] R.G. Payton, Elastic Wave Propagation in Transversely Isotropic Media, Nijhoff, The Hague, The Netherlands, 1983. [3] F. Honarvar, A.N. Sinclair, Nondestructive evaluation of cylindrical components by resonance acoustic spectroscopy, Ultrasonics 36 (1998) 845–854. [4] C.B. Scruby, L.E. Drain, Laser Ultrasonics: Techniques and Applications, Adam Hilger, New York, 1990. [5] Y. Pan, C. Rossignol, B. Audoin, Acoustic waves generated by a laser line pulse in a transversely isotropic cylinder, Appl. Phys. Lett. 82 (2003) 4379–4381. [6] Y. Pan, C. Rossignol, B. Audoin, Acoustic waves generated by a laser point source in an isotropic cylinder, J. Acoust. Soc. Am. 116 (2004) 814–820. [7] A. Rahman, F. Ahmad, Representation of the displacement in terms of scalar functions for use in transversely isotropic materials, J. Acoust. Soc. Am. 104 (1998) 3675–3676. [8] Y. Pan, C. Rossignol, B. Audoin, Identification of laser generated acoustic waves in the 2D transient response of cylinders, J. Acoust. Soc. Am. 117 (6) (2005) 3600–3608. [9] H.J. Maris, Effect of finite phonon wavelength on phonon focusing, Phys. Rev. B 28 (12) (1983) 7033–7037.