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Mass spring lattice modeling of the scanning laser source technique
Ultrasonics 39 (2002) 543–551 www.elsevier.com/locate/ultras
Mass spring lattice modeling of the scanning laser source technique Younghoon Sohn *, Sridhar Krishnaswamy Center for Quality Engineering and Failure Prevention, Northwestern University, 60208-3020 Evanston, IL, USA Accepted 28 January 2002
Abstract The scanning laser source (SLS) technique is a promising new laser ultrasonic tool for the detection of small surface-breaking defects. The SLS approach is based on monitoring the changes in laser generated ultrasound as a laser source is scanned over a defect. Changes in amplitude and frequency content have been observed for ultrasound generated by the laser over uniform and defective areas. In this paper, the SLS technique is simulated numerically using the mass spring lattice model. Thermoelastic laser generation of ultrasound in an elastic material is modeled using a shear dipole distribution. The spatial and temporal energy distribution profiles of typical pulsed laser sources are used to model the laser source. The amplitude and spectral variations in the laser generated ultrasound as the SLS scans over a large aluminum block containing a small surface-breaking crack are observed. The experimentally observed SLS amplitude and spectral signatures are shown to be captured very well by the model. In addition, the possibility of utilizing the SLS technique to size surface-breaking cracks that are sub-wavelength in depth is explored. Ó 2002 Published by Elsevier Science B.V. Keywords: Nondestructive testing; Laser ultrasonics; Scanning laser source technique; Mass spring lattice model; Surface-breaking cracks
1. Introduction Ultrasonic detection of surface-breaking cracks using Rayleigh waves is well established [1,2]. Conventionally such defects are detected using either pulse-echo or pitchcatch methods, where ultrasonic reflected or transmitted signals are monitored. Detection of surface-breaking flaws using laser-generated ultrasound is particularly attractive since it provides noncontact generation and detection of ultrasound with high spatial resolution [3]. Pulse-echo and pitch-catch methods can also be applied to laser ultrasonics if high power pulsed lasers are used to generate ultrasound [4,5]. However, when the depth of the crack is much smaller than the wavelength of the incident Rayleigh wave a significant portion of the Rayleigh wave energy passes right under the crack. This leads to quite small changes in observed signal amplitude (for both pulse-echo and pitch-catch methods) making it hard to detect the presence of such small defects.
*
Corresponding author.
The scanning laser source (SLS) technique is a promising new laser ultrasonic tool for the detection of small surface-breaking defects [6]. The SLS approach is based on monitoring the changes in laser generated ultrasound as a laser source is scanned over a defect. Changes in amplitude and frequency content are observed for ultrasound generated by the laser over uniform and defective areas. The SLS technique offers several advantages over conventional techniques. Ultrasonic generators such as contact PZT-transducers and near-contact electromagnetic acoustic transducers are not as easily scanned as a laser beam. Noncontact laser scanning can be easily done on specimens such as turbine disks which have complex geometries [7]. The SLS technique is not as sensitive as pulse-echo techniques to the orientation of the crack with respect to the generating/receiving transducer. Furthermore, when laser-based techniques such as the SLS use a fairly large generation source, it has been noted that they are less sensitive to grain noise, thereby enabling detection of small fatigue cracks [8]. Characteristic SLS signatures in terms of amplitude and spectral content variations have been observed as the generating laser scans over uniform and defective
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areas [6]. These changes have been attributed to both near-field scattering and to changes in the laser generation constraints in uncracked and cracked regions. In this paper, we describe a model to simulate the observed SLS signatures in order to help understand the SLS behavior and to aid in optimizing the inspection setup. The model used incorporates two features: (i) the laser ultrasonic source is modeled using a shear-dipole distribution on the surface of a structure, and (ii) the subsequent propagation and scattering of elastodynamic waves in the structure are modeled using a mass spring lattice. Of the many numerical methods that allow the simulation and analysis of elastic waves in various situations of interest to nondestructive evaluation [9], we use the mass spring lattice model (MSLM) in view of its excellent performance in simulation of elastic wave propagation both within the elastic media as well as on the surface [10,11]. The MSLM which is composed of lumped mass particles and springs represents the physical behavior of elastic media very well. In this paper, we first describe the SLS technique in detail. Details of the MSLM are then briefly given. It is shown that laser generation of ultrasound on surfaces with no defects is adequately modeled by the MSLM using distributed shear-dipoles. Finally, simulations of the SLS technique for various geometries are provided and compared with available experimental results. 2. Scanning laser source technique In the SLS technique, the ultrasound generation source, which is a point or a line-focused high-power laser beam, is swept across the test specimen surface and passes over surface-breaking flaws (see Fig. 1). The generated ultrasonic wave packet is detected using an optical interferometer either at a fixed location on the specimen or at a fixed distance between the source and receiver. The ultrasonic signal that arrives at the Rayleigh wave speed is monitored as the SLS is scanned. It is found that the amplitude and frequency of the measured ultrasonic signal have specific variations when the laser source approaches, passes over and moves behind the
Fig. 1. Schematic of the SLS technique.
Fig. 2. Typical characteristic signature of ultrasonic amplitude versus the SLS location as the source is scanned over a defect.
defect. Kromine et al. [6] have experimentally verified the SLS technique on an aluminum specimen with a surface-breaking fatigue crack of 4 mm length and 50 lm width. A broadband heterodyne interferometer with 1–15 MHz bandwidth was used as the ultrasonic detector. The SLS was formed by focusing a pulsed Nd:YAG laser beam (pulse duration––10 ns, energy––3 mJ) into a line of 5 mm length and 0.4 mm width (full width of the Gaussian at half the maximum). A plot of the amplitude of the detected ultrasonic signal versus the SLS position as the latter was scanned over the crack is presented in Fig. 2. The following aspects of this signature should be noted: (1) In the absence of a defect or when the source is far ahead of the defect, the amplitude of the generated ultrasonic direct signal is constant (see zone I in Fig. 2). The signal is of sufficient amplitude above the noise floor to be easily measured by the laser detector (see Fig. 3(a)). (2) As the source approaches the defect, the amplitude of the detected signal significantly increases (zone II in Fig. 2). This increase is readily detectable even with a low sensitivity laser interferometer as compared to weak echoes from the flaw (see Fig. 3(b)). The variation in signal amplitude is due to two mechanisms: (a) interaction of the direct ultrasonic wave with the waves scattered by the defect, and (b) changes in the conditions of generation of ultrasound when the SLS is in the vicinity of the defect. (3) As the source moves behind the defect, the amplitude drops lower than in zone I due to scattering of the generated signal by the defect (see zone III on a Fig. 2). In addition to the amplitude signature shown above, spectral variations in the detected ultrasonic signal also show characteristic features [6]. Both amplitude and spectral variations can form the basis for an inspection procedure using the SLS technique. These aspects are now explored using the MSLM simulations.
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Fig. 3. Representative ultrasonic time-domain signals detected by the heterodyne interferometer (at a fixed source to receiver distance) when the laser source is: (a) far ahead, (b) close to, and (c) behind the defect.
3. Mass spring lattice model for elastodynamics Elastodynamic wave propagation of laser generated ultrasound is simulated using the MSLM [10,11]. Only the plane strain problem for an isotropic elastic material is considered, and therefore only longitudinal and vertically-polarized transverse waves exist in the x–y plane of the structure. The elastic material has been modeled as a lattice of equidistantly placed rigid particles with elastic springs connecting the particles to each other (Fig. 4). The particles represent lumped masses corresponding to the dashed-rectangular regions shown. The linear and torsional springs represent the elastic stiffnesses of the material. The governing equations for the nodal displacements (of the rigid particles) in the horizontal (x) and vertical (y) directions are calculated by using a temporal and spatial difference formulation of the equations of motion for the center mass (i, j):
2 k1 k qðukþ1 i;j þ ui;j 2ui;j Þ=ðDtÞ
¼ Fx þ k1 ðukiþ1;j þ uki1;j 2uki;j Þ=h2 þ k2 ðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 þ uki1;j1 4uki;j Þ=2h2 þ k2 bðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 þ uki1;j1 4uki;j Þ=2h2 þ k2 ðvkiþ1;jþ1 vkiþ1;j1 vki1;jþ1 þ vki1;j1 Þ=2h2 þ k2 bðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 vki1;j1 Þ=2h2
ð1Þ 2 k1 k qðvkþ1 i;j þ vi;j 2vi;j Þ=ðDtÞ
¼ Fy þ k3 ðvki;jþ1 þ vki;j1 2vki;j Þ=h2 þ k2 ðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 þ vki1;j1 4vki;j Þ=2h2 þ k2 bðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 þ vki1;j1 4vki;j Þ=2h2 þ k2 ðukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þ uki1;j1 Þ=2h2 þ k2 bðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 uki1;j1 Þ=2h2 : ð2Þ
Fig. 4. Schematic of the MSLM.
Here h and Dt are the lattice grid spacing and the time kþ1 step respectively; ukþ1 i;j , vi;j denote the x and y direction displacements of the mass particle at the position (i, j) and at the time t ¼ kDt. Fx and Fy are the body force in the x and y directions, respectively; q is mass density of material; and k1 ; k2 ; k3 ; a and b are constants related to the elastic spring stiffnesses [10]. The body forces, Fx and Fy are in general given, and the displacements at previk1 k k ous time steps, uk1 i;j , vi;j , ui;j , or vi;j are known either from initial conditions or from the results of a prior time kþ1 step. Unknown displacements ukþ1 i;j , vi;j at the next time step are then calculated using Eqs. (1) and (2) explicitly. According to the Neumann analysis, the optimal ratio of the grid spacing h to time step Dt to obtain least phase error and the fastest computation speed [10] is given by the longitudinal (fastest) wave speed in the material:
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h ¼ Dt
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l q
ð3Þ
where k and l are Lame constants of the material. One of the advantages of the MSLM over other numerical methods is the ease of modeling planar free boundary surfaces and traction free planar crack surfaces by just disconnecting all the corresponding springs in the model and halving the masses on the free surface. In addition, it has been established that MSLM displacement fields associated with Rayleigh waves show quantitatively good agreement with analytical results [10]. The MSLM can be used to model wave propagation in finite-sized objects to monitor boundary reflections. For large-sized objects (or half spaces) it is effective to focus on a small region of interest and to use absorbing boundaries [11] to prevent spurious boundary reflections. The MSLM is therefore suitable for simulation of the SLS technique which inherently deals with the propagation and near-field scattering of Rayleigh waves.
4. Distributed shear-dipole model for thermoelastic laser generation of ultrasound When a laser pulse impinges on a solid surface one of the significant mechanisms that occurs is that electromagnetic energy of the laser source is converted into heat source [12]. The heat source leads to rapid thermal expansion causing ultrasonic wave generation. In this study, we shall restrict attention to the so-called thermoelastic regime where the laser-induced temperature rise is below the melting point of the test material, so that there is no ablation or plasma formation. For simplicity it is assumed that no significant optical penetration occurs below the surface of the test material. The absorption depth for 1064 nm laser wavelength on metallic materials such as aluminum or titanium (on which the SLS technique is most commonly applied) is in the nanometer range. Furthermore, thermal diffusion is neglected as it is not a significant factor within the time scale of interest. It is well established that a thermoelastic source at a point in the interior of an isotropic material can be modeled as three orthogonal force dipoles of equal strength [13]. The dipole strengths are proportional to the laser induced temperature rise (assuming no heat conductivity) [3]. For the case of a point laser source impinging on a free surface, a surface center of expansion approach has been proposed [14]. It has been shown that the surface center of expansion is equivalent to a pair of orthogonal dipoles parallel to the surface. For the case of an infinitesimally thin line laser source impinging on a surface, the surface center of expansion
essentially becomes a single shear dipole parallel to the surface. For a spatially distributed laser source, the model can be extended to include a spatially distributed set of dipoles. As the magnitude of the shear dipole is proportional to the temperature variation, the spatial and temporal temperature distribution across the surface is given by that of the laser source in the absence of heat conduction. Assuming a broad laser line source, the dipole magnitude Dðx; tÞ at location x and time t is given by: Dðx; tÞ ¼ AGðxÞqðtÞ
ð4Þ
Here A is a constant factor that includes the efficiency of conversion of electomagnetic to thermal energy along with elastic and thermal properties of the material, GðxÞ is the spatial energy density distribution and qðtÞ denotes the temporal pulse shape of the laser source. The laser output is assumed to have a Gaussian spatial distribution of the form GðxÞ ¼ eðxmÞ
2
=2r2
ð5Þ
where r is standard deviation and m is mean value of the distribution. This distribution of dipoles is applied in its discretized form in the MSLM as shown in Fig. 5. The full width of the Gaussian distribution (the 6r value) has been chosen as 420 lm based on the values typically used in experiments. In this case, the full width at half the maximum of the Gaussian C (C ¼ 2:355r) is then 165 lm. In the simulation, a grid spacing of h ¼ 14 lm is used to obtain 30 nodes over the 420 lm full width of the laser source. The corresponding time step needed to ensure minimal error satisfying stability condition [10] is given by Eq. (3) as Dt ¼ 2:37 ns. The temporal pulse shape for a Q-switched laser can be approximated by [15]. qðtÞ ¼ ðt=sÞeð1t=sÞ
ð6Þ
Fig. 5. Discretized laser source for simulation of Gaussian distribution.
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Fig. 6. Temporal profile of pulsed laser source.
where s is the pulse duration of the laser source. The Qswitched laser source pulse rise time has been chosen as 10 ns and the pulse lasts until about 60 ns. While this is longer than typically used in experiments, the bulk of the energy is within the first 20 ns of the pulse, which is close to the experimental value. Fig. 6 shows the temporal profile used in the model. For the MSLM calculation the body forces Fx at the nodes within the laser heated area are proportional to Dðx; tÞ in Eq. (1). In order to establish that this discretized dipole distribution adequately models thermoelastic laser generation of ultrasound, a test case of an uncracked aluminum block is considered. For the simulation 800 400 nodes (in x and y direction respectively) are used. The upper edge of the material on which the source is applied is constrained by traction free boundary conditions and absorbing boundaries are placed at the right, left and bottom edges to represent a half space. The material properties used in the simulation are: aluminum density q ¼ 2600 kg/m3 , and Lame constants k ¼ 4:1 1010 N/m2 , l ¼ 2:5 1010 N/m2 . Fig. 7 shows the displacements on the surface at different locations to the right of the excitation laser source (0.7, 1.4, and 2.8 mm in Fig. 7(a)–(c) respectively). The vertical and horizontal displacements are represented by solid and dashed lines respectively. Both longitudinal waves and Rayleigh waves can be clearly seen. It is found that the temporal variations observed using the MSLM are in good agreement with established analytical results [16]. Furthermore, the center frequency of the Rayleigh wave generated by a spatially Gaussian and temporally Dirac delta pulsed laser source is given by pffiffiffi 2CR fmax ¼ ð7Þ pd where CR is the Rayleigh wave speed and d is the width of Gaussian distribution, defined as the width where the
Fig. 7. Vertical (solid line) and horizontal (dashed line) displacements on the surface at various source-to-receiver distances ((a) 0.7 mm, (b) 1.4 mm, (c) 2.8 mm).
intensity drops to 1/e of the maximum value [17]. In the numerical simulation in Fig. 7 the frequency of the generated Rayleigh wave is measured to be 6.2 MHz, which is close (given that the pulse used here is not a Dirac delta) to the analytical result of 6.57 MHz calculated by Eq. (7). As a final check of the adequacy of the distributed dipole MSLM simulation of a thermoelastic laser source, the directivities of the longitudinal and shear bulk waves generated into the interior of the aluminum block are shown in Fig. 8(a) and (b) respectively. These results are found to agree well with analytically calculated results [18].
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Fig. 9. Vertical displacements at various SLS positions. Peak-to-peak vertical displacement (normalized).
Fig. 8. Calculated directivity pattern of laser generated ultrasound.
5. Modeling of the SLS technique for surface-breaking cracks Having established that the distributed dipole MSLM simulation of a thermoelastic laser source adequately captures the analytically and experimentally observed behavior, the MSLM is now used in the presence of surface-breaking cracks. The schematic of the SLS simulation using the MSLM is shown in Fig. 9(a). A surface-breaking crack of depth 28 lm is considered in a large aluminum block. The lattice spacing used is 14 lm. The SLS of full width 420 lm is modeled by a distribution of dipoles as before. The laser source is scanned across the specimen surface (and over the surface-breaking crack) in steps of 28 lm. Since a pair of nodes is needed to describe the equal and opposite forces that form a dipole, the scan step size is chosen to be double the nodal distance. This is necessary in order to keep the crack between pairs of dipoles rather than between a dipole nodal pair as the SLS is stepped. The effect of this is that the two nodal masses on either side of the crack are free to expand unconstrained towards each other, and this is quite similar to what is expected in reality. In this manner, the changed generation constraints when the SLS is over a crack is taken into account in the model. The calculated vertical displacements at a receiver location (fixed at 2.4 mm from the crack) for various
positions of the SLS are shown in Fig. 9(b). The SLS position is defined as the distance between the receiver and the center point of the laser source width. When the laser source is far ahead of the crack (positions 1, 2), direct and crack-reflected waves can be seen clearly in the time interval shown. When the laser source passes over the surface-breaking defect (positions 3–6), significant amplitude and spectral changes occur in the signal expected at the direct (from source to receiver ‘‘as the crow flies’’) Rayleigh wave arrival time. As the laser source approaches the crack (positions 3, 4), the vertical displacement significantly increases. When the SLS is directly over the crack (positions 4, 5), a trailing ultrasonic ringing is seen and this is attributed to Rayleigh waves that travel up and down the crack surface. Such ringing is not seen in experiments because real cracks are not mathematically smooth and will therefore scatter these signals away. As the source passes behind the crack (positions 5–7), the signal amplitude decreases due to screening by the crack. 5.1. SLS amplitude signature The experimentally measured SLS amplitude signatures are obtained using the direct Rayleigh wave arrivals. Therefore, a plot of the peak-to-peak amplitude of the vertical displacement (at the expected direct Rayleigh wave arrival time) versus the SLS position for the 28 lm crack depth is shown Fig. 10. When the source is far ahead of the crack (region A) the dominant waves are incident waves which directly propagate to the receiver without interacting with the crack (the reflected
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Fig. 10. Peak-to-peak vertical displacement versus SLS position.
waves which arrive later are relatively smaller than the incident waves). Thus the peak-to-peak vertical displacements show stable behavior and are seen to be the same as in the absence of a crack. As the source approaches and is over the crack, a significant increase of peak-to-peak vertical displacement occurs (region B) and in fact reaches its maximum when the center of the source is located just ahead of the crack. Since the energy of laser source has a Gaussian distribution, the greatest portion of the energy is at center of the source. Thus it is reasonable that the maximum occurs when the center of the source is just ahead of the crack since in this case the surface waves having the highest portion of energy are generated and these interact with the crack. As the source passes over the crack the amplitude decreases noticeably and becomes stable when the source is far behind the crack (region C). The crack depth chosen (28 lm) is much smaller than the center wavelength of the surface wave generated (466 lm), so that the greater portion of the incident energy just transmits past the crack. Therefore the peak-to-peak amplitudes in regions A and C show little difference. Note that the experimentally observed SLS amplitude signature (Fig. 2) and the MSLM predicted SLS signature are in good qualitative agreement with each other. 5.2. SLS spectral signature Kromine et al. [6] have also monitored characteristic spectral changes of the SLS signal where they note that the center frequency increases and subsequently decreases as the SLS scans a surface-breaking crack. Using the MSLM calculated vertical displacements for the various SLS positions shown in Fig. 9, the spectral content of the signals (window-gated around the direct Rayleigh wave arrival) were obtained using standard FFT processing. The center (maximum) frequency is plotted in Fig. 11 as a function of the SLS position, and
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Fig. 11. Maximum frequency versus SLS position.
it confirms the experimentally observed behavior. When the source is far ahead of the crack (region A), as one would expect, the maximum frequency (6.2 MHz) is the same as that of the Rayleigh wave in the absence of any cracks. The maximum frequency of surface waves generated by a laser source is inversely related to the width of the laser source as given by Eq. (7). When the SLS straddles the crack, the effective width of the laser source is reduced and this causes a shift to higher frequencies (region B). When the SLS is behind the crack (region C), the maximum frequencies drop and become stable. This is also expected from the fact that higher frequency components of the generated wave are better screened by the crack than lower frequency components. In the case shown, the difference in maximum frequency between regions A (far ahead) and C (far behind) is not very large since the crack depth is small compared to the Rayleigh wavelength. 5.3. SLS signatures for various crack depths The question of whether the SLS technique can be used to gauge crack depths is explored next. Fig. 12 shows the SLS amplitude signature for various crack depths. All the traces show the same general behavior described above. It is easy to detect the existence and location of a crack from the maximum of the SLS amplitude signature. However, there does not appear to be a strong correlation between the crack depth and the overall SLS amplitude maxima. However, when the source is far behind the crack, the amplitudes decrease as the crack depth gets larger. This is expected in that deeper cracks block more wave energy propagating along the surface and this is in fact the basis for pitchcatch techniques. A better picture emerges if only the amplitude of the 6.2 MHz component of the Rayleigh wave (corresponding to the center frequency of the generated
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Fig. 12. Peak-to-peak vertical displacement versus the SLS position for various crack depth.
wavepacket in the absence of a crack) is monitored. Fig. 13 shows the amplitude of the 6.2 MHz component wave versus SLS position for various crack depths. The differences in the maximum values are clearer than in Fig. 12. The amplitudes of the 6.2 MHz component wave versus crack depth are plotted in Fig. 14 for two SLS to receiver positions: 2.71 mm (just ahead of the crack) and 5.11 mm (well behind the crack). The amplitudes are normalized with respect to the Rayleigh wave amplitude obtained in the absence of a crack. The crack depth is normalized by the Rayleigh wavelength in the material at 6.2 MHz. The curve corresponding to the 5.11 mm SLS position simply indicates that the transmittivity of the crack decreases with crack depth. Of more significance is the curve corresponding to the 2.71 mm SLS position. This indicates that the SLS amplitude signature at a specified frequency increases rapidly with crack depth especially for small crack depths. Fig. 14 can serve as a calibration curve in experiments to size cracks that are sub-wavelength in depth.
Fig. 14. Normalized vertical displacement amplitude of the 6.2 MHz component wave versus normalized crack depth at specific SLS positions.
6. Conclusion Laser generation of Rayleigh waves has been modeled using distributed shear-dipoles and a MSLM. Good agreement with analytical results for time domain displacements, spectral content, and directivity were obtained for the case of generation on a half-space with no cracks. The model was then extended to the case of a half space with a crack in order to simulate the SLS technique. SLS amplitude and spectral signatures observed in SLS experiments have been reproduced by the model. Finally, a method to size sub-wavelength surface-breaking cracks has been described. Acknowledgement The authors are grateful to the Federal Aviation Administration’s Airworthiness Assurance Center of Excellence for their support. References
Fig. 13. Vertical displacements of 6.2 MHz components versus the SLS position for various crack depth.
[1] J. Krautkramer, H. Krautkramer, Ultrasonic Testing of Materials, 4th ed., Springer-Verlag, Berlin, 1990. [2] I.A. Viktorov, Rayleigh and Lamb waves, Plenum Press, New York, 1967. [3] C.B. Scruby, L.E. Drain, Laser Ultrasonics: Techniques and Applications, Adam Hilger, New York, 1990. [4] J.A. Cooper, R.J. Dewhurst, S.B. Palmer, Characterization of surface-breaking defects in metals with the use of laser-generated ultrasound, Philos. Trans. Roy. Soc. Lond., Math. Phys. Sci. A (1986) 319–320. [5] J. Huang, S. Krishnaswamy, J.D. Achenbach, Laser generation of narrow-band surface waves, J. Acoust. Soc. Am. 92 (5) (1992) 2527. [6] A.K. Kromine, P.A. Fomitchov, S. Krishnaswamy, J.D. Achenbach, Laser ultrasonic detection of surface-breaking discontinuities: Scanning laser source technique, Mater. Eval. 58 (2) (2000) 173. [7] P.A. Fomitchov, A.K. Kromine, Y. Sohn, S. Krishnaswamy, J.D. Achenbach, Ultrasonic imaging of small surface-breaking defects
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[8]
[9]
[10]
[11]
using scanning laser source technique, Review of Progress in Quantitative Nondestructive Evaluation, vol. 21A, Plenum Press, New York, 2002, p. 356. Z. Yan, P.B. Nagy, in: Enhanced laser generation of surface acoustic waves by discontinuities, Review of Progress in Quantitative Nondestructive Evaluation, vol. 20, Plenum Press, New York, 2000, p. 204. L.J. Bond, Numerical techniques and their use to study wave propagation and scattering: A review, in: S.K. Datta, J.D. Achenbach, Y.S. Rajapakse (Eds.), Elastic Waves and Ultrasonic Nondestructive Evaluation, North-Holland, Netherlands, 1990, p. 17. H. Yim, Y. Sohn, Numerical simulation and visualization of elastic waves using mass-spring lattice model, IEEE Trans., UFFC 47 (3) (2000) 549. H. Yim, Y. Choi, Simulation of ultrasonic waves in various type of elastic media using the mass spring lattice model, Mater. Eval. 58 (7) (2000) 889.
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[12] R.M. White, Generation of elastic waves by transient surface heating, J. Appl. Phys. 34 (12) (1963) 3559. [13] K. Aki, P.G. Richards, Quantitative Seismology, W.H. Freeman, San Francisco, 1980. [14] L.R.F. Rose, Point-source representation for laser-generated ultrasound, J. Acoust. Soc. Am. 75 (3) (1984) 723. [15] J.F. Ready, Effects of High-power Laser Radiation, Academic, New York, 1971, Chapter 3. [16] P.A. Doyle, Calculation of ultrasonic surface waves from an extended thermoelastic laser source, J. Nondestruct. Eval. 8 (3) (1989) 147. [17] Y.H. Berthelot, J. Jarzynski, in: Directional laser generation and detection of ultrasound with arrays of optical fivers, Review of Progress in Quantitative Nondestructive Evaluation, vol. 9, Plenum, New York, 1990, p. 463. [18] J.R. Bernstein, J.B. Spicer, Line source representation for lasergenerated ultrasound in aluminum, J. Acoust. Soc. Am. 107 (3) (2000) 1352.
Mass spring lattice modeling of the scanning laser source technique Younghoon Sohn *, Sridhar Krishnaswamy Center for Quality Engineering and Failure Prevention, Northwestern University, 60208-3020 Evanston, IL, USA Accepted 28 January 2002
Abstract The scanning laser source (SLS) technique is a promising new laser ultrasonic tool for the detection of small surface-breaking defects. The SLS approach is based on monitoring the changes in laser generated ultrasound as a laser source is scanned over a defect. Changes in amplitude and frequency content have been observed for ultrasound generated by the laser over uniform and defective areas. In this paper, the SLS technique is simulated numerically using the mass spring lattice model. Thermoelastic laser generation of ultrasound in an elastic material is modeled using a shear dipole distribution. The spatial and temporal energy distribution profiles of typical pulsed laser sources are used to model the laser source. The amplitude and spectral variations in the laser generated ultrasound as the SLS scans over a large aluminum block containing a small surface-breaking crack are observed. The experimentally observed SLS amplitude and spectral signatures are shown to be captured very well by the model. In addition, the possibility of utilizing the SLS technique to size surface-breaking cracks that are sub-wavelength in depth is explored. Ó 2002 Published by Elsevier Science B.V. Keywords: Nondestructive testing; Laser ultrasonics; Scanning laser source technique; Mass spring lattice model; Surface-breaking cracks
1. Introduction Ultrasonic detection of surface-breaking cracks using Rayleigh waves is well established [1,2]. Conventionally such defects are detected using either pulse-echo or pitchcatch methods, where ultrasonic reflected or transmitted signals are monitored. Detection of surface-breaking flaws using laser-generated ultrasound is particularly attractive since it provides noncontact generation and detection of ultrasound with high spatial resolution [3]. Pulse-echo and pitch-catch methods can also be applied to laser ultrasonics if high power pulsed lasers are used to generate ultrasound [4,5]. However, when the depth of the crack is much smaller than the wavelength of the incident Rayleigh wave a significant portion of the Rayleigh wave energy passes right under the crack. This leads to quite small changes in observed signal amplitude (for both pulse-echo and pitch-catch methods) making it hard to detect the presence of such small defects.
*
Corresponding author.
The scanning laser source (SLS) technique is a promising new laser ultrasonic tool for the detection of small surface-breaking defects [6]. The SLS approach is based on monitoring the changes in laser generated ultrasound as a laser source is scanned over a defect. Changes in amplitude and frequency content are observed for ultrasound generated by the laser over uniform and defective areas. The SLS technique offers several advantages over conventional techniques. Ultrasonic generators such as contact PZT-transducers and near-contact electromagnetic acoustic transducers are not as easily scanned as a laser beam. Noncontact laser scanning can be easily done on specimens such as turbine disks which have complex geometries [7]. The SLS technique is not as sensitive as pulse-echo techniques to the orientation of the crack with respect to the generating/receiving transducer. Furthermore, when laser-based techniques such as the SLS use a fairly large generation source, it has been noted that they are less sensitive to grain noise, thereby enabling detection of small fatigue cracks [8]. Characteristic SLS signatures in terms of amplitude and spectral content variations have been observed as the generating laser scans over uniform and defective
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areas [6]. These changes have been attributed to both near-field scattering and to changes in the laser generation constraints in uncracked and cracked regions. In this paper, we describe a model to simulate the observed SLS signatures in order to help understand the SLS behavior and to aid in optimizing the inspection setup. The model used incorporates two features: (i) the laser ultrasonic source is modeled using a shear-dipole distribution on the surface of a structure, and (ii) the subsequent propagation and scattering of elastodynamic waves in the structure are modeled using a mass spring lattice. Of the many numerical methods that allow the simulation and analysis of elastic waves in various situations of interest to nondestructive evaluation [9], we use the mass spring lattice model (MSLM) in view of its excellent performance in simulation of elastic wave propagation both within the elastic media as well as on the surface [10,11]. The MSLM which is composed of lumped mass particles and springs represents the physical behavior of elastic media very well. In this paper, we first describe the SLS technique in detail. Details of the MSLM are then briefly given. It is shown that laser generation of ultrasound on surfaces with no defects is adequately modeled by the MSLM using distributed shear-dipoles. Finally, simulations of the SLS technique for various geometries are provided and compared with available experimental results. 2. Scanning laser source technique In the SLS technique, the ultrasound generation source, which is a point or a line-focused high-power laser beam, is swept across the test specimen surface and passes over surface-breaking flaws (see Fig. 1). The generated ultrasonic wave packet is detected using an optical interferometer either at a fixed location on the specimen or at a fixed distance between the source and receiver. The ultrasonic signal that arrives at the Rayleigh wave speed is monitored as the SLS is scanned. It is found that the amplitude and frequency of the measured ultrasonic signal have specific variations when the laser source approaches, passes over and moves behind the
Fig. 1. Schematic of the SLS technique.
Fig. 2. Typical characteristic signature of ultrasonic amplitude versus the SLS location as the source is scanned over a defect.
defect. Kromine et al. [6] have experimentally verified the SLS technique on an aluminum specimen with a surface-breaking fatigue crack of 4 mm length and 50 lm width. A broadband heterodyne interferometer with 1–15 MHz bandwidth was used as the ultrasonic detector. The SLS was formed by focusing a pulsed Nd:YAG laser beam (pulse duration––10 ns, energy––3 mJ) into a line of 5 mm length and 0.4 mm width (full width of the Gaussian at half the maximum). A plot of the amplitude of the detected ultrasonic signal versus the SLS position as the latter was scanned over the crack is presented in Fig. 2. The following aspects of this signature should be noted: (1) In the absence of a defect or when the source is far ahead of the defect, the amplitude of the generated ultrasonic direct signal is constant (see zone I in Fig. 2). The signal is of sufficient amplitude above the noise floor to be easily measured by the laser detector (see Fig. 3(a)). (2) As the source approaches the defect, the amplitude of the detected signal significantly increases (zone II in Fig. 2). This increase is readily detectable even with a low sensitivity laser interferometer as compared to weak echoes from the flaw (see Fig. 3(b)). The variation in signal amplitude is due to two mechanisms: (a) interaction of the direct ultrasonic wave with the waves scattered by the defect, and (b) changes in the conditions of generation of ultrasound when the SLS is in the vicinity of the defect. (3) As the source moves behind the defect, the amplitude drops lower than in zone I due to scattering of the generated signal by the defect (see zone III on a Fig. 2). In addition to the amplitude signature shown above, spectral variations in the detected ultrasonic signal also show characteristic features [6]. Both amplitude and spectral variations can form the basis for an inspection procedure using the SLS technique. These aspects are now explored using the MSLM simulations.
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Fig. 3. Representative ultrasonic time-domain signals detected by the heterodyne interferometer (at a fixed source to receiver distance) when the laser source is: (a) far ahead, (b) close to, and (c) behind the defect.
3. Mass spring lattice model for elastodynamics Elastodynamic wave propagation of laser generated ultrasound is simulated using the MSLM [10,11]. Only the plane strain problem for an isotropic elastic material is considered, and therefore only longitudinal and vertically-polarized transverse waves exist in the x–y plane of the structure. The elastic material has been modeled as a lattice of equidistantly placed rigid particles with elastic springs connecting the particles to each other (Fig. 4). The particles represent lumped masses corresponding to the dashed-rectangular regions shown. The linear and torsional springs represent the elastic stiffnesses of the material. The governing equations for the nodal displacements (of the rigid particles) in the horizontal (x) and vertical (y) directions are calculated by using a temporal and spatial difference formulation of the equations of motion for the center mass (i, j):
2 k1 k qðukþ1 i;j þ ui;j 2ui;j Þ=ðDtÞ
¼ Fx þ k1 ðukiþ1;j þ uki1;j 2uki;j Þ=h2 þ k2 ðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 þ uki1;j1 4uki;j Þ=2h2 þ k2 bðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 þ uki1;j1 4uki;j Þ=2h2 þ k2 ðvkiþ1;jþ1 vkiþ1;j1 vki1;jþ1 þ vki1;j1 Þ=2h2 þ k2 bðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 vki1;j1 Þ=2h2
ð1Þ 2 k1 k qðvkþ1 i;j þ vi;j 2vi;j Þ=ðDtÞ
¼ Fy þ k3 ðvki;jþ1 þ vki;j1 2vki;j Þ=h2 þ k2 ðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 þ vki1;j1 4vki;j Þ=2h2 þ k2 bðvkiþ1;jþ1 þ vkiþ1;j1 þ vki1;jþ1 þ vki1;j1 4vki;j Þ=2h2 þ k2 ðukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þ uki1;j1 Þ=2h2 þ k2 bðukiþ1;jþ1 þ ukiþ1;j1 þ uki1;jþ1 uki1;j1 Þ=2h2 : ð2Þ
Fig. 4. Schematic of the MSLM.
Here h and Dt are the lattice grid spacing and the time kþ1 step respectively; ukþ1 i;j , vi;j denote the x and y direction displacements of the mass particle at the position (i, j) and at the time t ¼ kDt. Fx and Fy are the body force in the x and y directions, respectively; q is mass density of material; and k1 ; k2 ; k3 ; a and b are constants related to the elastic spring stiffnesses [10]. The body forces, Fx and Fy are in general given, and the displacements at previk1 k k ous time steps, uk1 i;j , vi;j , ui;j , or vi;j are known either from initial conditions or from the results of a prior time kþ1 step. Unknown displacements ukþ1 i;j , vi;j at the next time step are then calculated using Eqs. (1) and (2) explicitly. According to the Neumann analysis, the optimal ratio of the grid spacing h to time step Dt to obtain least phase error and the fastest computation speed [10] is given by the longitudinal (fastest) wave speed in the material:
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h ¼ Dt
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l q
ð3Þ
where k and l are Lame constants of the material. One of the advantages of the MSLM over other numerical methods is the ease of modeling planar free boundary surfaces and traction free planar crack surfaces by just disconnecting all the corresponding springs in the model and halving the masses on the free surface. In addition, it has been established that MSLM displacement fields associated with Rayleigh waves show quantitatively good agreement with analytical results [10]. The MSLM can be used to model wave propagation in finite-sized objects to monitor boundary reflections. For large-sized objects (or half spaces) it is effective to focus on a small region of interest and to use absorbing boundaries [11] to prevent spurious boundary reflections. The MSLM is therefore suitable for simulation of the SLS technique which inherently deals with the propagation and near-field scattering of Rayleigh waves.
4. Distributed shear-dipole model for thermoelastic laser generation of ultrasound When a laser pulse impinges on a solid surface one of the significant mechanisms that occurs is that electromagnetic energy of the laser source is converted into heat source [12]. The heat source leads to rapid thermal expansion causing ultrasonic wave generation. In this study, we shall restrict attention to the so-called thermoelastic regime where the laser-induced temperature rise is below the melting point of the test material, so that there is no ablation or plasma formation. For simplicity it is assumed that no significant optical penetration occurs below the surface of the test material. The absorption depth for 1064 nm laser wavelength on metallic materials such as aluminum or titanium (on which the SLS technique is most commonly applied) is in the nanometer range. Furthermore, thermal diffusion is neglected as it is not a significant factor within the time scale of interest. It is well established that a thermoelastic source at a point in the interior of an isotropic material can be modeled as three orthogonal force dipoles of equal strength [13]. The dipole strengths are proportional to the laser induced temperature rise (assuming no heat conductivity) [3]. For the case of a point laser source impinging on a free surface, a surface center of expansion approach has been proposed [14]. It has been shown that the surface center of expansion is equivalent to a pair of orthogonal dipoles parallel to the surface. For the case of an infinitesimally thin line laser source impinging on a surface, the surface center of expansion
essentially becomes a single shear dipole parallel to the surface. For a spatially distributed laser source, the model can be extended to include a spatially distributed set of dipoles. As the magnitude of the shear dipole is proportional to the temperature variation, the spatial and temporal temperature distribution across the surface is given by that of the laser source in the absence of heat conduction. Assuming a broad laser line source, the dipole magnitude Dðx; tÞ at location x and time t is given by: Dðx; tÞ ¼ AGðxÞqðtÞ
ð4Þ
Here A is a constant factor that includes the efficiency of conversion of electomagnetic to thermal energy along with elastic and thermal properties of the material, GðxÞ is the spatial energy density distribution and qðtÞ denotes the temporal pulse shape of the laser source. The laser output is assumed to have a Gaussian spatial distribution of the form GðxÞ ¼ eðxmÞ
2
=2r2
ð5Þ
where r is standard deviation and m is mean value of the distribution. This distribution of dipoles is applied in its discretized form in the MSLM as shown in Fig. 5. The full width of the Gaussian distribution (the 6r value) has been chosen as 420 lm based on the values typically used in experiments. In this case, the full width at half the maximum of the Gaussian C (C ¼ 2:355r) is then 165 lm. In the simulation, a grid spacing of h ¼ 14 lm is used to obtain 30 nodes over the 420 lm full width of the laser source. The corresponding time step needed to ensure minimal error satisfying stability condition [10] is given by Eq. (3) as Dt ¼ 2:37 ns. The temporal pulse shape for a Q-switched laser can be approximated by [15]. qðtÞ ¼ ðt=sÞeð1t=sÞ
ð6Þ
Fig. 5. Discretized laser source for simulation of Gaussian distribution.
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Fig. 6. Temporal profile of pulsed laser source.
where s is the pulse duration of the laser source. The Qswitched laser source pulse rise time has been chosen as 10 ns and the pulse lasts until about 60 ns. While this is longer than typically used in experiments, the bulk of the energy is within the first 20 ns of the pulse, which is close to the experimental value. Fig. 6 shows the temporal profile used in the model. For the MSLM calculation the body forces Fx at the nodes within the laser heated area are proportional to Dðx; tÞ in Eq. (1). In order to establish that this discretized dipole distribution adequately models thermoelastic laser generation of ultrasound, a test case of an uncracked aluminum block is considered. For the simulation 800 400 nodes (in x and y direction respectively) are used. The upper edge of the material on which the source is applied is constrained by traction free boundary conditions and absorbing boundaries are placed at the right, left and bottom edges to represent a half space. The material properties used in the simulation are: aluminum density q ¼ 2600 kg/m3 , and Lame constants k ¼ 4:1 1010 N/m2 , l ¼ 2:5 1010 N/m2 . Fig. 7 shows the displacements on the surface at different locations to the right of the excitation laser source (0.7, 1.4, and 2.8 mm in Fig. 7(a)–(c) respectively). The vertical and horizontal displacements are represented by solid and dashed lines respectively. Both longitudinal waves and Rayleigh waves can be clearly seen. It is found that the temporal variations observed using the MSLM are in good agreement with established analytical results [16]. Furthermore, the center frequency of the Rayleigh wave generated by a spatially Gaussian and temporally Dirac delta pulsed laser source is given by pffiffiffi 2CR fmax ¼ ð7Þ pd where CR is the Rayleigh wave speed and d is the width of Gaussian distribution, defined as the width where the
Fig. 7. Vertical (solid line) and horizontal (dashed line) displacements on the surface at various source-to-receiver distances ((a) 0.7 mm, (b) 1.4 mm, (c) 2.8 mm).
intensity drops to 1/e of the maximum value [17]. In the numerical simulation in Fig. 7 the frequency of the generated Rayleigh wave is measured to be 6.2 MHz, which is close (given that the pulse used here is not a Dirac delta) to the analytical result of 6.57 MHz calculated by Eq. (7). As a final check of the adequacy of the distributed dipole MSLM simulation of a thermoelastic laser source, the directivities of the longitudinal and shear bulk waves generated into the interior of the aluminum block are shown in Fig. 8(a) and (b) respectively. These results are found to agree well with analytically calculated results [18].
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Fig. 9. Vertical displacements at various SLS positions. Peak-to-peak vertical displacement (normalized).
Fig. 8. Calculated directivity pattern of laser generated ultrasound.
5. Modeling of the SLS technique for surface-breaking cracks Having established that the distributed dipole MSLM simulation of a thermoelastic laser source adequately captures the analytically and experimentally observed behavior, the MSLM is now used in the presence of surface-breaking cracks. The schematic of the SLS simulation using the MSLM is shown in Fig. 9(a). A surface-breaking crack of depth 28 lm is considered in a large aluminum block. The lattice spacing used is 14 lm. The SLS of full width 420 lm is modeled by a distribution of dipoles as before. The laser source is scanned across the specimen surface (and over the surface-breaking crack) in steps of 28 lm. Since a pair of nodes is needed to describe the equal and opposite forces that form a dipole, the scan step size is chosen to be double the nodal distance. This is necessary in order to keep the crack between pairs of dipoles rather than between a dipole nodal pair as the SLS is stepped. The effect of this is that the two nodal masses on either side of the crack are free to expand unconstrained towards each other, and this is quite similar to what is expected in reality. In this manner, the changed generation constraints when the SLS is over a crack is taken into account in the model. The calculated vertical displacements at a receiver location (fixed at 2.4 mm from the crack) for various
positions of the SLS are shown in Fig. 9(b). The SLS position is defined as the distance between the receiver and the center point of the laser source width. When the laser source is far ahead of the crack (positions 1, 2), direct and crack-reflected waves can be seen clearly in the time interval shown. When the laser source passes over the surface-breaking defect (positions 3–6), significant amplitude and spectral changes occur in the signal expected at the direct (from source to receiver ‘‘as the crow flies’’) Rayleigh wave arrival time. As the laser source approaches the crack (positions 3, 4), the vertical displacement significantly increases. When the SLS is directly over the crack (positions 4, 5), a trailing ultrasonic ringing is seen and this is attributed to Rayleigh waves that travel up and down the crack surface. Such ringing is not seen in experiments because real cracks are not mathematically smooth and will therefore scatter these signals away. As the source passes behind the crack (positions 5–7), the signal amplitude decreases due to screening by the crack. 5.1. SLS amplitude signature The experimentally measured SLS amplitude signatures are obtained using the direct Rayleigh wave arrivals. Therefore, a plot of the peak-to-peak amplitude of the vertical displacement (at the expected direct Rayleigh wave arrival time) versus the SLS position for the 28 lm crack depth is shown Fig. 10. When the source is far ahead of the crack (region A) the dominant waves are incident waves which directly propagate to the receiver without interacting with the crack (the reflected
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Fig. 10. Peak-to-peak vertical displacement versus SLS position.
waves which arrive later are relatively smaller than the incident waves). Thus the peak-to-peak vertical displacements show stable behavior and are seen to be the same as in the absence of a crack. As the source approaches and is over the crack, a significant increase of peak-to-peak vertical displacement occurs (region B) and in fact reaches its maximum when the center of the source is located just ahead of the crack. Since the energy of laser source has a Gaussian distribution, the greatest portion of the energy is at center of the source. Thus it is reasonable that the maximum occurs when the center of the source is just ahead of the crack since in this case the surface waves having the highest portion of energy are generated and these interact with the crack. As the source passes over the crack the amplitude decreases noticeably and becomes stable when the source is far behind the crack (region C). The crack depth chosen (28 lm) is much smaller than the center wavelength of the surface wave generated (466 lm), so that the greater portion of the incident energy just transmits past the crack. Therefore the peak-to-peak amplitudes in regions A and C show little difference. Note that the experimentally observed SLS amplitude signature (Fig. 2) and the MSLM predicted SLS signature are in good qualitative agreement with each other. 5.2. SLS spectral signature Kromine et al. [6] have also monitored characteristic spectral changes of the SLS signal where they note that the center frequency increases and subsequently decreases as the SLS scans a surface-breaking crack. Using the MSLM calculated vertical displacements for the various SLS positions shown in Fig. 9, the spectral content of the signals (window-gated around the direct Rayleigh wave arrival) were obtained using standard FFT processing. The center (maximum) frequency is plotted in Fig. 11 as a function of the SLS position, and
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Fig. 11. Maximum frequency versus SLS position.
it confirms the experimentally observed behavior. When the source is far ahead of the crack (region A), as one would expect, the maximum frequency (6.2 MHz) is the same as that of the Rayleigh wave in the absence of any cracks. The maximum frequency of surface waves generated by a laser source is inversely related to the width of the laser source as given by Eq. (7). When the SLS straddles the crack, the effective width of the laser source is reduced and this causes a shift to higher frequencies (region B). When the SLS is behind the crack (region C), the maximum frequencies drop and become stable. This is also expected from the fact that higher frequency components of the generated wave are better screened by the crack than lower frequency components. In the case shown, the difference in maximum frequency between regions A (far ahead) and C (far behind) is not very large since the crack depth is small compared to the Rayleigh wavelength. 5.3. SLS signatures for various crack depths The question of whether the SLS technique can be used to gauge crack depths is explored next. Fig. 12 shows the SLS amplitude signature for various crack depths. All the traces show the same general behavior described above. It is easy to detect the existence and location of a crack from the maximum of the SLS amplitude signature. However, there does not appear to be a strong correlation between the crack depth and the overall SLS amplitude maxima. However, when the source is far behind the crack, the amplitudes decrease as the crack depth gets larger. This is expected in that deeper cracks block more wave energy propagating along the surface and this is in fact the basis for pitchcatch techniques. A better picture emerges if only the amplitude of the 6.2 MHz component of the Rayleigh wave (corresponding to the center frequency of the generated
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Fig. 12. Peak-to-peak vertical displacement versus the SLS position for various crack depth.
wavepacket in the absence of a crack) is monitored. Fig. 13 shows the amplitude of the 6.2 MHz component wave versus SLS position for various crack depths. The differences in the maximum values are clearer than in Fig. 12. The amplitudes of the 6.2 MHz component wave versus crack depth are plotted in Fig. 14 for two SLS to receiver positions: 2.71 mm (just ahead of the crack) and 5.11 mm (well behind the crack). The amplitudes are normalized with respect to the Rayleigh wave amplitude obtained in the absence of a crack. The crack depth is normalized by the Rayleigh wavelength in the material at 6.2 MHz. The curve corresponding to the 5.11 mm SLS position simply indicates that the transmittivity of the crack decreases with crack depth. Of more significance is the curve corresponding to the 2.71 mm SLS position. This indicates that the SLS amplitude signature at a specified frequency increases rapidly with crack depth especially for small crack depths. Fig. 14 can serve as a calibration curve in experiments to size cracks that are sub-wavelength in depth.
Fig. 14. Normalized vertical displacement amplitude of the 6.2 MHz component wave versus normalized crack depth at specific SLS positions.
6. Conclusion Laser generation of Rayleigh waves has been modeled using distributed shear-dipoles and a MSLM. Good agreement with analytical results for time domain displacements, spectral content, and directivity were obtained for the case of generation on a half-space with no cracks. The model was then extended to the case of a half space with a crack in order to simulate the SLS technique. SLS amplitude and spectral signatures observed in SLS experiments have been reproduced by the model. Finally, a method to size sub-wavelength surface-breaking cracks has been described. Acknowledgement The authors are grateful to the Federal Aviation Administration’s Airworthiness Assurance Center of Excellence for their support. References
Fig. 13. Vertical displacements of 6.2 MHz components versus the SLS position for various crack depth.
[1] J. Krautkramer, H. Krautkramer, Ultrasonic Testing of Materials, 4th ed., Springer-Verlag, Berlin, 1990. [2] I.A. Viktorov, Rayleigh and Lamb waves, Plenum Press, New York, 1967. [3] C.B. Scruby, L.E. Drain, Laser Ultrasonics: Techniques and Applications, Adam Hilger, New York, 1990. [4] J.A. Cooper, R.J. Dewhurst, S.B. Palmer, Characterization of surface-breaking defects in metals with the use of laser-generated ultrasound, Philos. Trans. Roy. Soc. Lond., Math. Phys. Sci. A (1986) 319–320. [5] J. Huang, S. Krishnaswamy, J.D. Achenbach, Laser generation of narrow-band surface waves, J. Acoust. Soc. Am. 92 (5) (1992) 2527. [6] A.K. Kromine, P.A. Fomitchov, S. Krishnaswamy, J.D. Achenbach, Laser ultrasonic detection of surface-breaking discontinuities: Scanning laser source technique, Mater. Eval. 58 (2) (2000) 173. [7] P.A. Fomitchov, A.K. Kromine, Y. Sohn, S. Krishnaswamy, J.D. Achenbach, Ultrasonic imaging of small surface-breaking defects
Y. Sohn, S. Krishnaswamy / Ultrasonics 39 (2002) 543–551
[8]
[9]
[10]
[11]
using scanning laser source technique, Review of Progress in Quantitative Nondestructive Evaluation, vol. 21A, Plenum Press, New York, 2002, p. 356. Z. Yan, P.B. Nagy, in: Enhanced laser generation of surface acoustic waves by discontinuities, Review of Progress in Quantitative Nondestructive Evaluation, vol. 20, Plenum Press, New York, 2000, p. 204. L.J. Bond, Numerical techniques and their use to study wave propagation and scattering: A review, in: S.K. Datta, J.D. Achenbach, Y.S. Rajapakse (Eds.), Elastic Waves and Ultrasonic Nondestructive Evaluation, North-Holland, Netherlands, 1990, p. 17. H. Yim, Y. Sohn, Numerical simulation and visualization of elastic waves using mass-spring lattice model, IEEE Trans., UFFC 47 (3) (2000) 549. H. Yim, Y. Choi, Simulation of ultrasonic waves in various type of elastic media using the mass spring lattice model, Mater. Eval. 58 (7) (2000) 889.
551
[12] R.M. White, Generation of elastic waves by transient surface heating, J. Appl. Phys. 34 (12) (1963) 3559. [13] K. Aki, P.G. Richards, Quantitative Seismology, W.H. Freeman, San Francisco, 1980. [14] L.R.F. Rose, Point-source representation for laser-generated ultrasound, J. Acoust. Soc. Am. 75 (3) (1984) 723. [15] J.F. Ready, Effects of High-power Laser Radiation, Academic, New York, 1971, Chapter 3. [16] P.A. Doyle, Calculation of ultrasonic surface waves from an extended thermoelastic laser source, J. Nondestruct. Eval. 8 (3) (1989) 147. [17] Y.H. Berthelot, J. Jarzynski, in: Directional laser generation and detection of ultrasound with arrays of optical fivers, Review of Progress in Quantitative Nondestructive Evaluation, vol. 9, Plenum, New York, 1990, p. 463. [18] J.R. Bernstein, J.B. Spicer, Line source representation for lasergenerated ultrasound in aluminum, J. Acoust. Soc. Am. 107 (3) (2000) 1352.