SAW dispersion measurements for ultrasonic characterization of surface- treated metals

SAW dispersion measurements for ultrasonic characterization of surfacetreated metals Alberto Ruiz* - Peter B. Nagy* *Department of Aerospace Engineeri...
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SAW dispersion measurements for ultrasonic characterization of surfacetreated metals Alberto Ruiz* - Peter B. Nagy* *Department of Aerospace Engineering and Engineering Mechanics University of Cincinnati, Cincinnati, Ohio 43221-0070, USA [email protected], [email protected] Nondestructive evaluation of the prevailing compressive residual stress in the shallow subsurface layer of surface-treated metals is greatly complicated by the accompanying surface roughness and cold work. A high-precision laser-ultrasonic technique has been developed to study the feasibility of SAW dispersion spectroscopy for residual stress assessment on surface-treated metals in a wide frequency range from 1.5 MHz to 15 MHz. This paper presents experimental data obtained on samples that have been surface treated with shot peening, laser shock peening and low plasticity burnishing, which indicate that the dispersion of the surface wave is affected by surface roughness, compressive residual stress, and cold work. Although surface roughness induced scattering provides a significant contribution to the observed dispersion of the SAW, our experimental data indicate that it is feasible to observe a substantial and highly characteristic change in the velocity of the SAW when the specimen is heat treated at different annealing temperatures. These results also indicate that precise SAW dispersion measurements can be exploited to monitor thermal relaxation and recovery in surface-treated metals. ABSTRACT:

KEYWORDS:

Ultrasonic Surface Waves, Spectroscopy, Nondestructive Testing, Shot Peening.

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1. Introduction Surface wave dispersion measurements can be used to nondestructively characterize shot-peened, laser shock-peened, burnished, and otherwise surfacetreated metals. In recent years, there have been numerous efforts to separate the contribution of surface roughness, which is not affected by thermo-mechanical relaxation, from those of near-surface material variations, i.e., the primary residual stress effect and the secondary cold work effects (such as anisotropic texture and increased dislocation density), which all significantly decay during relaxation. Stateof-the-art laser-ultrasonic scanning and sophisticated digital signal processing methods allow us to measure the velocity on rough shot-peened specimens with high accuracy (Ruiz and Nagy, 2002). It was found that the measured dispersion of the surface wave arises from three different sources. First, there is an apparent dispersion due to the diffraction of the Surface Acoustic Wave (SAW) as it travels over the surface of the specimen. This dispersion effect is on the order of 0.1%, which is significantly higher than the experimental error associated with the measurement and comparable to the expected velocity change produced by nearsurface compressive residual stresses in metals below their yield point. Second, there is a real but spurious dispersion caused by SAW scattering on the rough surface, which is an adverse geometrical byproduct of certain surface treatment procedures such as shot peening. Finally, there is the principal dispersion caused by a number of material effects of the surface treatment, including the primary compressive residual stress effect and the secondary cold work effect. The cold work effect itself can be further divided into at least three main contributions, namely anisotropic texture, increased dislocation density, and grain refinement/coarsening. Generally, all these effects must be carefully taken into consideration and properly accounted for in order to quantitatively characterize the effect of surface treatment on the fatigue life of fracture-critical components.

2. Background Lord Rayleigh was the first to demonstrate that acoustic waves can propagate over the free surface of an elastic half-space. The amplitude of the particle displacement decays exponentially with depth and becomes negligible deep below the surface. The particles of the solid move along elliptical trajectories, with the major axis of the ellipse being normal to the surface. Rayleigh-type surface acoustic waves (SAWs) produce elastic stresses that are confined to the surface within a shallow depth of approximately one wavelength. Therefore, they are particularly sensitive to the boundary conditions at the surface and to the material properties in the near-surface region. Since the penetration depth of surface waves is frequencydependent, high-frequency SAWs are affected only by the material properties close to the surface while the low-frequency components give information on the bulk material properties of the substrate. This frequency-dependence can be exploited to

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control the depth of the near-surface region to be inspected. Another interesting feature of surface waves is that they are less affected by beam spreading than bulk waves, since they are confined to travel on the surface of the material, therefore they diverge only in two dimensions rather than in three like bulk modes. There are numerous classical studies in the literature on the relationship between SAW dispersion and gradients of various physical properties of the substrate (Tittman et al., 1974, Richardson, 1977, Richardson and Tittman, 1977, Tittman and Richardson, 1978, Hirao et al., 1981, Auld, 1990). The use of surface acoustic waves for the characterization of surface roughness, surface residual stress, and coating thickness measurements has been studied by several authors more recently (Lakestani et al., 1995, Ditri and Hongerholt, 1996, Lavrentyev et al., 1999, Glorieux and Gao, 2000, Delsanto et al., 2000, Duquennoy et al., 2001, Lavrentyev and Veronesi, 2001). In addition, the effect of surface cracks on the attenuation and dispersion of Rayleigh waves was explored in great depth using ultrasonic frequencies (Zhang and Achenbach, 1990, Warren et al., 1996, and Pecorari, 1996, 1998, 2000, and 2001) as well as Brillouin scattering (Medik and Sathish, 1994, Mutti et al., 1995). These studies have found that the total change in the Rayleigh wave velocity is typically less than 1%. One of the main challenges, of course, is to measure the Rayleigh wave velocity with sufficiently high accuracy. One method capable of measuring SAW velocity with such a high precision is scanning acoustic microscopy. However, scanning acoustic microscopy is very difficult to apply on the slightly rough and curved surfaces presented by typical surface treated components, therefore in the rest of this paper we are going to focus entirely on laser-ultrasonic measurements which allow us to measure the velocity on rough shot-peened specimens with 0.01-0.02% accuracy.

2.1. Laser interferometry Laser-ultrasonic methods are particularly well suited for SAW measurements (for a recent review see Lomonosov et al., 2001). Laser interferometry employs the principle of optical interference to recover the sought acoustic information from the light reflected from or scattered by a surface subject to ultrasonic vibration. It is a very sensitive way of measuring displacements and velocities, which requires a highly monochromatic light source and thus the use of lasers is virtually necessary. Interferometers used for the detection of ultrasonic movements of surfaces can be divided into two main types. In the first type, the light reflected from the surface is made to interfere with a reference beam, giving a measurement of optical phase difference and hence instantaneous surface displacement. The second type of interferometer is designed as a high-resolution optical spectrometer to detect the socalled Doppler shift in the frequency of the reflected light from a moving surface and gives an output dependent on the velocity of the surface. The first type is more widely used and more practical at lower frequencies while the second type offers potentially higher sensitivity for rough surfaces at high frequencies. Both types are non-contacting and do not disturb the ultrasonic field as contact devices do. This

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feature makes laser probing the preferred alternative to contact methods in investigating the spatial evolution of surface waves, their dispersion, diffraction and damping.

2.2. Shot peening Figure 1 shows a schematic representation of the shot peening process and the different penetration exhibited by its three major effects, namely residual stress, cold work, and surface roughness. During peening, a stream of round shots is aimed at the surface of the material. The impact of these shots introduces beneficial compressive residual stresses and other, mostly adverse, secondary effects, such as surface roughness and cold work. This process is widely used in the aerospace industry to increase the damage tolerance of metal parts since a thin surface layer of compressive residual stress could prevent crack initiation and retard crack growth during service. Unfortunately, the accompanying cold work significantly reduces the relaxation and recrystallization temperatures of the material, therefore accelerates the detrimental process of thermo-mechanical stress relief. This inherent instability of the beneficial near-surface compressive residual stress is the main reason why nondestructive evaluation of shot-peened metal surfaces is so important. In this paper we will demonstrate that ultrasonic evaluation of the compressive residual stress in the layer directly beneath the surface is greatly complicated by the previously mentioned adverse effects of shot peening.

residual stress

cold work

surface roughness

Figure 1. Main effects of shot peening including compressive residual stress, cold work, and surface roughness. The resulting residual stresses can be best assessed by the X-ray diffraction method, which measures changes in atomic inter-planar spacing to determine the magnitude of the prevailing elastic strain. This method is nondestructive and can be used to explore stresses locally only in a very shallow surface layer because of the

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poor penetration depth of X-rays in metals. Unfortunately, the residual stress so easily measured on the surface is rather sensitive to spurious effects and is not characteristic of the whole protective compressive layer below the surface. The Xray diffraction method can be expanded to depths below ≈0.05 mm only by destructive sectioning. The less conventional neutron diffraction method is based on the same physical principle as X-ray diffraction, but offers much better penetration depth. Among the biggest obstacles to employing neutron diffraction are bulkiness, expense, and hazards associated with neutron beam sources. Therefore, there is a continued need for quantitative nondestructive evaluation techniques capable of characterizing the magnitude and depth profile of subsurface residual stresses. Ultrasonic SAW dispersion measurements offer one of the most promising approaches to achieve this goal. In the following sections, we are going to discuss how the main effects of shot peening affect the propagation of acoustic surface waves on shot-peened surfaces.

2.3. Surface roughness effect In the past 30 years, the attention of numerous researchers has been focused on the investigation of surface acoustic waves propagating along the randomly rough surface of an elastic solid. It was found that on rough surfaces the surface acoustic wave exhibits scattering-induced attenuation and dispersion (Eguiluz and Maradudin, 1983, Krylov and Smirnova, 1990, Mayer and Lehner, 1993, Kosachev and Shchegrov, 1995). Figure 2 shows numerical calculations based on the meanfield perturbation theory (Eguiluz and Maradudin, 1983). The theoretically predicted roughness-induced dispersion is further illustrated in Fig. 3. The maximum dispersion for 10 µm rms roughness and 100 µm correlation length occurs at approximately 5 MHz, while for 7 µm rms roughness and 60 µm correlation length the point of maximum dispersion is at approximately 8 MHz. The initial lowfrequency behavior of the predicted SAW dispersion has been experimentally verified (Krylov and Smirnova, 1990), but the apparent minimum and subsequent maximum (not shown in this figure) have not been experimentally observed yet. It is important to mention that the relatively weak dispersion effect is accompanied by strong scattering-induced attenuation which adversely affects the accuracy of SAW velocity measurements. In Fig. 3, those parts of the dispersion curves, which exhibit less than 10 dB/cm attenuation coefficient, therefore are expected to be easily observed in our experiments, are highlighted in thick lines. The excess scattering-induced attenuation may be one reason why this phenomenon has not been observed experimentally yet. Most engineering surface enhancement processes produce a surface roughness that does not have the optimal correlation length to rms surface roughness ratio, which would push the dispersion maximum to lower frequencies and could facilitate the observation of the velocity minimum and possibly even the subsequent maximum at higher frequencies.

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Relative Velocity Change [%]

0

Correlation length:

-0.1

100 µm

-0.2 -0.3

80 µm

-0.4

70 µm

-0.5 60 µm

-0.6 -0.7 -0.8

50 µm

-0.9 -1 0

2

4

6

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10

Frequency [MHz] Figure 2. Theoretical predictions of the surface roughness induced dispersion for 10 µm rms roughness and different correlation lengths.

Relative Velocity Change [%]

0 -0.1 10 µm, 100 µm

-0.2 -0.3 7 µm, 60 µm

-0.4 rms roughness, correlation length

-0.5 0

5

10

15

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Frequency [MHz] Figure 3. Theoretical predictions for SAW dispersion on rough surfaces (thick solid lines represent low-scattering regions of less than 10 dB/cm attenuation coefficient).

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In previous experimental studies it was assumed that the dispersion of the surface wave was mainly caused by surface roughness, which is not affected by thermal relaxation (see, for example, Krylov and Smirnova, 1990). In contrast, our results in shot-peened aluminum show that gradual heat treatment in a protective atmosphere perceivably decreases the dispersion of the surface wave by gradually removing the residual stress and cold work effects. This substantial change in the observed SAW dispersion is caused by relaxation and recrystallization processes during heat treatment.

2.4. Residual stress and cold work effects During cold work the dislocation density grows very substantially, the metal becomes harder, a localized anisotropic texture is introduced, and strength properties significantly increase. Morphological distortion of the grain structure due to plastic deformation also occurs during cold work. Two of the main effects produced by shot peening, namely residual stress and cold work, can significantly change during service as a result of thermo-mechanical relaxation, which can be simulated under laboratory conditions by appropriate heat treatments that cause thermally activated stress release. In cold-worked metals, annealing even at modest temperatures can also initiate recrystallization, thereby reversing the effects of plastic deformation and restoring the earlier softer, more ductile, unworked grain structure. Generally, the annealing temperature has a stronger effect than time in producing microstructural changes during recrystallization. Strain energy, absorbed by the metal during deformation, is deployed in the stress field that surrounds the dislocations. This simultaneously strengthens the matrix and raises its free energy. Reduction of the dislocation density lowers the system's free energy during annealing and recrystallization. These processes generally require elevated temperatures and sufficient time for the atoms to diffuse; only then can dislocations move, annihilate, climb or disappear in larger numbers. It is customary to divide the response to annealing into three temperature regimes: recovery, recrystallization, and grain growth. The recovery stage (primary recrystallization) is characterized by a small drop in hardness and virtually no change in the grain structure. During this process a redistribution of dislocations occurs, although the density of dislocations is only slightly reduced leaving barriers to their motion mostly intact. As a result, the metal retains much of its high strength and low-ductility. Notably, however, the residual stress is eliminated or at least greatly reduced in this process. During recrystallization, more and more of the coldworked matrix transforms into new strain-free crystals. A more heavily cold-worked matrix will recrystallize at lower temperature. Furthermore, a high nucleation density caused by the presence of crystallographic imperfections implies that the recrystallized grain size will be finer. After the grains have recrystallized, they are

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stable but their size can further increase if the temperature is kept high or increased. As the grains grow, the matrix softens significantly and becomes even more ductile.

3. Experimental Method and Results Figure 4 shows a schematic diagram of the experimental arrangement used in our surface wave dispersion measurements (Ruiz and Nagy, 2002). A wedge transducer was mounted on the specimen and excited by a Panametrics 5270 broadband pulser. Three 6-mm-diameter Gamma-S Krautkramer screw-in transducers of 2.25, 5, and 10 MHz nominal center frequency were used on an W-048 90°-wedge also made by Krautkramer. The propagating surface wave was detected by a LUIS 35 Fabry-Perot interferometer. The laser beam was aimed at the surface of the specimen at a small angle of incidence and an objective lens focused the diffuse reflection on to the tip of an optical fiber connected to the interferometer. The ultrasonic signal detected by the interferometer was digitized and averaged by a LeCroy 9310 oscilloscope and then sent to a computer for further processing. The specimen was mounted on a Velmex translation table and the relative position between the wedge transmitter and the laser spot was changed by a computer accessible stepping motor controller. It should be mentioned that this technique is similar to previously developed laser-ultrasonic systems where a pulsed laser was used to excite the SAW and a broadband piezoelectric transducer was used to detect it (Schneider and Schwarz, 1997, Schneider et al., 2000) except that the role of the laser and piezoelectric transducer is exchanged. tracking Fabry-Perot interferometer

digital oscilloscope averaged signal

optical fiber computer trigger

objective lens

Nd:YAG laser

pulser

primary lens

wedge transducer

laser beam specimen

SAW translation table

stepping motor driver

Figure 4. Experimental setup for laser-ultrasonic SAW velocity measurement.

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Figure 5 shows the details of the scanning procedure used in our measurements, First, we aimed the laser spot at a 15-mm distance from the front of a wedge transducer and then scanned the beam along the propagation direction over a total distance of 30 mm, i.e., up to 45 mm from the wedge, in steps of 80 µm. The scanning resolution of the translation table was 3.175 µm. At each of the 300 axial positions, we averaged 1,000 pulses to increase the signal-to-noise ratio. In addition, during averaging we also used lateral scanning normal to the propagation direction to further improve the accuracy of the measurement. The lateral scanning dimension was ±1 mm, i.e., significantly smaller than the beam width (the diameter of the piezoelectric transducer mounted on the wedge was 6 mm). In this way, both temporal and spatial averaging was achieved at the same time and the laser interferometer effectively acted as a 2-mm-long line receiver. lateral scanning axial scanning wedge

transducer

laser beam treated surface SAW

Figure 5. A schematic of the scanning area to be inspected by the laser probe. Both the data acquisition and the scanner were controlled by the same LabView program. In order to assure the absolute accuracy of our velocity measurements, the temperature of the specimen was stabilized at 26.6 ºC within ±0.2 ºC (temperature variations are not expected to lead to dispersion unless the surface temperature of the specimen is significantly different from that of the interior). As described above, spatial averaging was achieved during lateral scanning normal to the wave propagation direction to reduce the incoherent scattering by (i) surface roughness and (ii) the inhomogeneous microstructure as well as (iii) coherent diffraction effects, especially in the near-field of the transmitter. A slower step-wise scanning was performed in the axial direction, i.e., parallel to the wave propagation, in order to map the phase of the surface vibration as a function of the propagation distance. The LabView program gradually changed the delay time of the LeCroy digital oscilloscope with respect to the trigger signal by using the programmable digital delay and highly accurate quartz master clock of the oscilloscope. At each step the new delay time was calculated from the scanning position using a nominal tracking

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velocity. In most cases, the recorded signal moved only very slightly either forward or backward within the data window depending on whether the chosen tracking velocity cn was a little higher or lower, respectively, than the actual surface wave velocity. In an ideal case, the tracking velocity is very close to the actual one, and the broadband pulse does not move appreciably at all, though its shape slightly changes as a result of dispersion. The recorded data was spectrum analyzed by a Discrete Fourier Transform algorithm that determined the phase of the signal at 10-20 different frequencies depending on the bandwidth of the transmitter (the bandwidth of the interferometer is essentially flat from 2 MHz to 100 MHz). In order to verify that the measurement system is sufficiently accurate to study the rather weak dispersion exhibited by ultrasonic surface waves propagating on surface-treated metals, first a couple of 2024-T351 aluminum bars of 50.8 mm (width) × 203 mm (length) × 12.7 mm (height) were carefully polished with 2,000grade sandpaper parallel to the length of the specimen to minimize scattering by surface irregularities. Then, the specimens were heat treated at 345 ºC for one hour. The objective of this initial heat treatment was to remove all near-surface material variations that might be present in the as-received cold-rolled bar stock material. These variations are associated with subtle near surface effects such as the presence of residual stresses, elevated hardness, increased dislocation density, and anisotropic texture. With the exception of surface roughness induced scattering, annealing above the recrystallization temperature of the material effectively eliminates all other nearsurface variations which could otherwise contribute to the observed surface wave dispersion. It should be mentioned that, to some degree, annealing also changes the microstructure and can lead to perceivable grain coarsening when it is overdone. As a result, the surface wave velocity might also change slightly, but, since the effect is essentially the same throughout the whole volume of the specimen, it does not necessarily lead to strong surface wave dispersion, though it might cause some microstructural dispersion in both bulk and surface wave propagation via grain scattering. After heat treatment, the Al2024-T351 samples were shot peened over a 50.8 mm × 50.8 mm square area at their center using 0.3-mm-diameter MI-110-H shots and 100 % coverage. Three series of specimens were prepared using 4A, 6A, and 10A Almen intensities (Eckersley and Champaigne, 1991). The unpeened smooth side of one of the specimens was used as a reference (0A). As an example, Fig. 6 shows the normalized phase as a function of the propagation distance obtained from a 4A-intensity shot-peened specimen using a 3.5-MHz transducer. In order to achieve the highest possible phase accuracy, we have to limit the overall range over which the phase might change. To a large degree, this is achieved by carefully selecting the nominal tracking velocity that is used to control the delay of the digitizer. However, some time shift will inevitably occur, which in turn will produce large changes in phase that are linearly proportional to frequency. These changes might hide smaller frequency-dependent changes caused by dispersive wave propagation, therefore must be further suppressed. This was achieved by “normalizing” the actual phase ϕ measured at any given frequency f to the center frequency fn of the transducer according to ϕ n = ϕ fn / f . This

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normalization of the phase allowed us to directly observe the change in the slope of the phase due to dispersion as well as to avoid folding back of the phase at 360°intervals.

Normalized Phase [10°/div]

4.7 MHz 4.5 MHz 4.3 MHz 4.1 MHz 3.9 MHz 3.7 MHz 3.5 MHz 3.3 MHz 3.1 MHz 2.9 MHz 2.7 MHz 2.5 MHz 2.3 MHz 2.1 MHz 1.9 MHz 1.7 MHz

30

40

50

60

70

80

90

Propagation Distance [mm] Figure 6. Example of the measured normalized phase versus propagation distance curves on an Almen 4 shot-peened specimen at sixteen different frequencies. The velocity of the SAW was calculated from the slope of the normalized phase using the following formula (Ruiz and Nagy, 2002) cm =

cn cn = , cn ∂ϕ cn ∂ϕ n 1+ 1+ ω ∂z ω n ∂z

[1]

where cm is the measured surface wave velocity, and cn is the tracking velocity chosen to minimize the observed total phase variation over the fairly long propagation distance. On the right side of Eq. [1] we introduced the angular frequency used for normalization ω n = 2 π fn , which is typically chosen to be either the nominal or actual center frequency of the transmitter. The average slope of the normalized phase was calculated at each frequency by finding the best fitting linear approximation to all data points using the least-mean-square method. In Figure 6, the low-frequency slopes are slightly negative, which, according with Eq. [1], means that the low-frequency components propagated at a velocity slightly higher than the tracking velocity. In comparison, the high-frequency slopes are positive,

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which indicates that they propagated slower than the tracking velocity, i.e., the velocity decreased with increasing frequency over the bandwidth of the transmitter. Figure 7 shows the results of our surface wave velocity measurements on three different aluminum specimens using three different transmitters of 2.25 MHz, 5 MHz and 10 MHz nominal center frequencies. Interestingly, a perceivable dispersion effect is observed even on the smooth unpeened side. This apparent dispersion effect is mainly caused by the diffraction of the slightly divergent SAW as it propagates over the surface of the specimen (Ruiz and Nagy, 2002). On the other hand, the two shot-peened specimens exhibit much stronger dispersion which increases with peening intensity. This effect is caused partly by surface roughness induced scattering and partly by the other previously mentioned material effects of shot peening. 2.96 2.95 2.94 2.93

smooth

2.92

Almen 6

2.91

Almen 10

2.9 2.89 2.88 0

5

10

15

20

Frequency [MHz]

Figure 7. Surface wave velocity measurements on untreated shot-peened aluminum specimens. 3.1. SAW dispersion on heat-treated shot-peened specimens Heat treating the sample at different temperatures allows us to observe the effect of annealing temperature and time on the surface wave velocity by gradually removing the effect of cold work and residual stress. Therefore, a series of subsequent heat treatments were conducted on the shot-peened specimens, starting from 150 ºC up to 325 ºC in steps of 25 ºC. The specimens were held for one hour at each temperature in a protective nitrogen atmosphere. After each heat treatment, the velocity of the surface acoustic wave was measured again for the same frequency range on the treated and smooth sides of the sample. The objective of measuring the velocity on the smooth side was to verify that the surface wave velocity does not change significantly as a result of heat treatment.

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Figure 8 shows the results of the velocity measurements on an Almen 10 sample after a series of heat treatments. The SAW velocity on the unpeened smooth side does not change significantly during the heat treatments and the single data set shown in this figure is the average of the eight individual measurements. These results illustrate that heat treatment at 325 ºC reduces the dispersion by roughly 2/3 compared to the untreated condition. Also, it can be observed that initially most of the velocity change occurs at high frequencies. As the heat treatment temperature increases, the dispersion becomes gradually weaker at high frequencies while continues a little longer at lower frequencies.

2.96

smooth

2.95

325 °C

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300 °C

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275 °C 250 °C

2.92

225 °C

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200 °C

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175 °C

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150 °C untreated

2.88 0

5

10

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Frequency [MHz] Figure 8. Surface wave velocity measurements on a shot-peened (Almen 10) aluminum specimen after different levels of thermal relaxation. Typically, the penetration depth of the excess cold work is only one-third of the penetration depth of the residual stress. Therefore, the thermally induced relaxation and recrystallization processes occur much faster directly below the surface than at larger depths. As a result, initially the near-surface region is much more affected than the material at larger depths, which explains why the low-frequency components of the interrogating SAW are initially much less sensitive to annealing than the highfrequency components. As the temperature increases, the relative velocity difference between the peened and unpeened specimens further decreases at low frequencies, which indicates that the relaxation continues at larger depths. However, the velocity does not change any further at high frequencies, which indicates that very close to the surface the relaxation and recrystallization has been essentially completed. Figure 9 shows the same results relative to the baseline velocity spectrum measured on the unpeened reference surface. To obtain these curves, for every heat

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treatment level we subtracted the average phase velocity measured over the smooth part of the specimen, which includes the effects of diffraction and grain scattering induced dispersion, from the data obtained over the peened part. By doing this, it is possible to better observe the combined effects of surface roughness, residual stress and cold work on the SAW velocity. The relative velocity curves gradually approach the expected level of the surface roughness induced dispersion, which was calculated using the mean-field perturbation approach (Eguiluz and Maradudin, 1983, Kosachev and Shchegrov, 1995). In the weak roughness approximation, this method can yield analytical solutions for both the attenuation and dispersion of the surface wave. In these calculations, the rms roughness and correlation length of the shotpeened surface was assumed to be 7.5 µm and 60 µm, respectively, which are consistent with white light interference microscopic measurements. Above 325 ºC heat treatment temperature (not shown in Figs. 8 and 9) we observed that the velocity difference continues to decrease a little further below 10 MHz, but at higher frequencies it begins to slightly increase again after reaching a minimum. At this relatively high temperature the near-surface region of the sample might have been completely recrystallized and grain coarsening might have started, which could explain the additional dispersion at high frequencies.

Relative Velocity Change [%] a

0 -0.2

roughness

-0.4

325 °C

-0.6

300 °C

-0.8

275 °C 250 °C

-1

225 °C

-1.2

200 °C

-1.4

175 °C

-1.6

150 °C

-1.8

untreated

-2 0

5

10

15

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Frequency [MHz] Figure 9. Relative SAW velocity on a 10A shot-peened aluminum specimen after increasing levels of thermal relaxation at eight different temperatures. Theoretical predictions for the surface roughness induced dispersion are also shown (dotted line, 7.5 µm rms roughness and 60 µm correlation length).

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Similar measurements were also conducted on an aluminum specimen shotpeened to a lower 6 Almen intensity. Figure 10 shows the dispersion curves obtained on this specimen before annealing as well as after a series of heat treatments. We can observe that the dispersion is smaller than in the Almen 10 sample and the over all change upon heat treatment is only ≈0.01 mm/µs. Figure 11 shows the relative change in velocity for this specimen. The untreated data shows the total effect of shot peening on the velocity. After heat treating the sample at different temperatures, the relative velocity change gradually decreases up to about 225 ºC, above which no significant further relaxation is observed at high frequencies. At low frequencies, the dispersion further decreases as a result of continuing partial relaxation and recrystallization deep below the surface, but at 6 MHz the velocity versus frequency curve exhibits a fairly flat plateau region up to 10 MHz were it starts to decrease again due to the emerging grain scattering. Increasing dispersion at very high frequencies after an initial decay as the specimen is repeatedly heat treated at increasing temperatures has been observed in numerous cases. The most probable cause of this behavior is accelerated grain coarsening in the heavily cold worked near-surface layer of the specimen, although conclusive evidence to support this conclusion is not available yet at this time.

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Velocity [mm/µs]

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smooth

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225 °C 200 °C

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150 °C untreated

2.9 2.89 2.88 0

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Frequency [MHz] Figure 10. Surface wave velocity measurements on an Almen 6 shot-peened aluminum specimen heat treated at different temperatures.

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0 -0.1 -0.2 -0.3 -0.4

300 °C

-0.5

225 °C

-0.6

200 °C

-0.7

175 °C

-0.8

150 °C

-0.9

untreated

-1 0

5

10

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Frequency [MHz] Figure 11. Relative SAW velocity measurements on an Almen 6 shot-peened aluminum specimen heat treated at different temperatures.

3.2. Acoustoelastic effect The above results clearly indicate that the thermal relaxation process significantly reduces the dispersion of ultrasonic surface waves on shot-peened aluminum surfaces. To quantify how much of this change is actually due to residual stress relaxation and how much to the recrystallization of the cold-worked microstructure, it would be necessary to measure the residual stress and cold work by X-ray diffraction at each heat treatment temperature. At this point, such measurements have not been done yet, therefore we do not know the exact behavior of residual stress and cold work during heat treatment, although coarse qualitative estimates are possible based on recently published results in ductile copper (Carreon et al., 2002). However, even without X-ray diffraction results, we can ascertain that the presence of compressive near-surface residual stresses should increase rather than decrease the SAW velocity, therefore the measured dispersion must be dominated by the various effects of cold work. The relative velocity change β of the surface wave propagating at a given angle θ to the principal stress direction can be calculated as follows (Lavrentyev et al., 1999):

SAW dispersion measurements

β =

c − c0 1 1 = ( K ⊥ + K|| ) (σ1 + σ 2 ) + ( K ⊥ − K|| ) (σ1 − σ2 ) cos(2θ) , c0 2 2

17

[2]

where c and c0 are the sound velocities in the stressed and unstressed states, respectively, K|| and K⊥ are the acoustoelastic constants for parallel and normal orientations, respectively, and σ1 and σ2 are the principal stresses. For shotpeened surfaces the principal stresses are approximately equal (σ1 = σ 2 = σ ) and Eq. [2] can be simplified as follows

β = ( K ⊥ + K|| ) σ .

[3]

A series of experiments were conducted in a 2024-T351 aluminum specimen to determine the crucial acoustoelastic constants of the material. For simplicity, the coefficients were estimated from simple shear-wave birefringence measurements, which can be considered as approximations for the corresponding SAW coefficients. We measured the stress-dependence of the shear velocities normal to uniaxial tension and compression with polarization both normal and parallel to the load. Figure 12 shows a plot of the experimental results for the change of the relative velocity with tensile stress after correcting for the Poisson effect. The acoustoelastic constants K⊥= 7.63×10-5 ksi-1 and K|| = -15.3×10-5 ksi-1 were calculated from linear regression of the data. 0.4 0.3 0.2 0.1 0 -0.1 parallel polarization normal polarization

-0.2 -0.3 -20

-10

0

10

20

Axial Stress [ksi]

Figure 12. Relative shear wave velocity change as function of uniaxial stress.

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Knowing that the maximum residual stress that can be introduced below the surface by shot peening is comparable to the yield stress σY, it is possible to estimate the maximum relative change in the velocity as:

βY = − ( K ⊥ + K|| ) σY ,

[4]

where βY is the maximum relative change in velocity at the maximum compressive stress, which is equal to the yield strength of the material, σY. Using σY ≈ 50 ksi for Al2024-T351, βY was found to be approximately 0.4 % and positive. However, the SAW measurements revealed an overall negative dispersion of about 1 %, which clearly shows that the residual stress effect is not dominant in the dispersion of the surface wave. This conclusion is consistent with earlier findings for aluminum alloys (Lavrentyev et al., 1999, Glorieux and Gao, 2000).

3.3. Cold work effects The previous results indicate that the cause of the negative dispersion, which decays during heat treatment, must be attributed to texture and increased dislocation density produced by the shot peening process. At this point, there is not much known about the localized texture produced by shot peening directly below the surface in Al2024. However, circumstantial evidence indicates that it might play a significant, possibly even dominant, role in the observed SAW dispersion. One related problem, that we know much more about and can study experimentally much easier, is the texture in cold-rolled plates. Such plates are typically orthotropic with the three principal directions oriented along the rolling or length direction (L), the transverse or width direction (W), and the thickness direction (T). Because of symmetry requirements, the velocity of the shear wave propagating in the length direction with polarization along the width direction (cLW) is the same as that of the wave propagating in the width direction with polarization along the length direction (cWL), i.e. cLW = cWL. Similarly, cLT = cTL and cTW = cWT. Based on shear wave velocity measurements in cold-rolled Al2024-T351 plates, cLW > cTW > cLT and cLW is typically ≈0.4% higher, while cTW and cLT are ≈0.1% and ≈0.3% lower, respectively, than the isotropic average. In other words, the type of plastic deformation caused by rolling of aluminum plates is expected to reduce the surface wave by η ≈ 0.2%. Needless to say that without tempering the induced anisotropy would be much stronger. Furthermore, the amount of maximum plastic deformation directly below the surface of shot-peened specimens could be much stronger than the levels of plastic deformation found in rolled plates, therefore the surface wave velocity could drop by as much η ≈ 1-2%. In order to assess the effect of texture on surface wave dispersion on shot-peened surfaces, beside the magnitude of the anisotropy, we also need to know something about its depth distribution. X-ray diffraction measurements indicate that the depthdependence of the cold work is roughly exponential with penetration depth δ that is

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19

roughly 1/3 of the correlation length of the surface roughness which can be relatively easily measured (Carreon et al., 2002). All these rough estimates can be combined into a rudimentary model to predict the resulting surface wave dispersion. Figure 13 shows the calculated surface wave dispersion for η = 2% and four different penetration depths δ = 50, 100, 200, and 400 µm. In each case the depth profile was discretized in ten layers over a 2δ total depth and the dispersion curve was calculated by Disperse 2 (Pavlakovic and Lowe, 2000). Of course, more sophisticated approaches based on multi-layered anisotropic models (Nayfeh, 1995) could be readily used to obtain more accurate predictions for the cold work induced surface wave dispersion along shot-peened surfaces, but the added computational difficulties could not be justified by the limited purpose of this illustration. These results suggest that texture induced surface wave dispersion is probably the dominant factor in the observed total dispersion on shot-peened aluminum surfaces.

0 δ= 50 µm -1

100 µm 200 µm 400 µm

-2 0

5

10

15

20

Frequency [MHz] Figure 13. Calculated surface wave dispersion at different penetration depths for relative anisotropy of η = 2%. The surface acoustic wave dispersion caused by anisotropic texture is negative, i.e., the velocity decreases with increasing frequency, its spectral behavior is consistent with the observed dispersion in shot-peened specimens, and it is stronger at higher Almen intensities mostly because of the deeper penetration depth of cold work. Qualitatively, the texture induced dispersion is rather similar to the surface roughness scattering induced dispersion, but it does vanish with annealing while the surface roughness effect lingers even after full relaxation and recrystallization. Among the various effects of shot peening, in unrelaxed specimens cold work induced texture appears to play the most important role in the dispersion of surface acoustic waves. However, the other major effect of cold work, namely increased

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dislocation density, should be also investigated as it might be a significant contributing factor. Depending on the level of the cold work present in the material, stress release starts in aluminum alloys at approximately 130 °C. We detected a perceivable change in the SAW velocity at 150 °C, which indicates that the change in the dispersion curves measured by the laser-ultrasonic method can be used to indirectly detect early relaxation of the residual stress in shot-peened specimens. As a result of this thermal relaxation, the negative texture-induced dispersion gradually decreases, along with the positive residual stress effect, therefore the relative contribution of the also negative but unaffected surface roughness effect increases. Because of this, in order to quantitatively assess the level of thermo-mechanical relaxation in the material, the surface roughness induced dispersion must be properly corrected for, especially after partial relief.

3.4. Other surface treatment methods Mechanical surface enhancement methods differ in how the surface is deformed and in the magnitude and depth profile of the resulting residual stress and cold work. Low-Plasticity Burnishing (LPB) is a surface enhancement method that produces a deep layer of high compression with improved surface finishing and minimal cold work, which greatly reduces the unwanted thermo-mechanical relaxation of desirable compressive stresses. LPB is made by a smooth free-rolling spherical or cylindrical tool in a single pass with a normal force just sufficient to deform the material to the desired depth, creating a compressive layer of residual stress with a controlled amount of cold working. The magnitude of compression produced by LPB at the surface is comparable to that of other surface enhancement processes, but the depth of the compressive layer is larger by as much as one order of magnitude. Another surface enhancement method is Laser Shock Peening (LSP), where a laser pulse is focused onto the surface of the part which has been covered with a tape that provides an ablative medium. Laser irradiation vaporizes a small portion of this layer, which creates an explosive pressure. A compressive residual stress is produced when the maximum stress of the shock wave exceeds the dynamic yield strength of the metal. Using water as a physical constraint generates a pressure in the range of 0.9-1.5 × 106 psi (6-10 GPa), which is far more than adequate to plastically yield the metal surface and create the beneficial compressive residual stresses. The depth of the induced compressive stress depends on the attenuation of the shock wave, which is material dependent. The affected depth typically ranges from 0.5 to 1.5 mm. The induced compressive stress decreases with increasing distance from the surface. Harder materials tend to develop deeper affected depths. Multiple laser pulses can be used to drive the compressive stresses deeper below the surface by work hardening the metal surface during each pulse, lowering the attenuation of the shock wave. By comparison, the residual stress developed using traditional shot peening is limited to a depth of approximately 0.25 mm.

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In the following demonstration, we used five samples of IN100 nickel-base superalloy. Two of them were shot peened at 6A and 8A intensities, one sample was laser-shock peened and the two remaining samples were low-plasticity burnished at two different levels. The surface wave velocity was measured on each sample's untreated and treated surfaces. Figure 14 shows the Rayleigh wave velocity as a function of frequency on the shot-peened samples. As in the aluminum samples, the dispersion increases more or less linearly with frequency. Figure 15 shows the results of similar measurements on the laser-shock peened specimen, where the observed dispersion is perceivably weaker. Finally, Fig. 16 shows the measured SAW dispersion in two different grades of low-plasticity burnished samples. In this case the dispersion is significant even at low frequencies because the penetration depth of the residual stress and texture is relatively deep. At the same time, the dispersion at high frequencies does not increase significantly since these samples do not exhibit significant surface roughness.

Velocity [mm/µs]

3.06

3.04 untreated 3.02

shot-peened A lm en 6 shot-peened A lm en 8

3 0

5

10

15

Frequency [M Hz]

Figure 14. Ultrasonic SAW velocity versus frequency in shot-peened IN100 nickelbase superalloy specimens. In order to better observe the effect of these surface treatments on the Rayleigh wave dispersion, we subtracted the velocity of the untreated area from that of the treated one. By doing this, we removed all other effects (diffraction, grain scattering induced dispersion, etc.) and the results represent only the effects of the surface treatment, i.e., surface roughness, residual stress and cold work. Figure 17 shows the normalized dispersion curves for all three types of surface treatment. Laser-shock peening produces similar, but weaker dispersion than low-plasticity burnishing. This is because the large penetration depth results in a plateau region over the upper part of the frequency range. We can also see for shot peening that, at high frequencies, SAWs are more affected by the material properties close to the surface and by surface topography. Surface roughness induced dispersion more or less linearly

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increases with frequency and becomes a dominating factor at the high end of the spectrum.

Velocity [mm/µs]

3

2.98

2.96

untreated laser-shocked

2.94 0

5

10

15

Frequency [M H z]

Figure 15. Ultrasonic SAW velocity versus frequency in laser-shock peened IN100 nickel-base superalloy specimens.

Velocity [mm/µs]

3.06

3.04

untreated

3.02

low-grade LPB high-grade LPB

3 0

5

10

15

Frequency [M H z]

Figure 16. Ultrasonic SAW velocity versus frequency in low-plasticity burnished IN100 nickel-base superalloy specimens.

SAW dispersion measurements

23

0 -0.2 -0.4 -0.6 laser-shocked

-0.8

shot peened Almen 6

-1

shot peened Almen 8 low-grade LPB

-1.2

high-grade LPB -1.4 0

5

10

15

Frequency [MHz] Figure 17. Normalized velocity change as a function of frequency in different surface-treated IN100 nickel-base superalloy specimens.

4. Conclusions Surface wave dispersion measurements are often used in ultrasonic nondestructive testing to characterize near-surface material variations that might lead to significant dispersion. In addition to actual dispersion, a perceivable dispersion effect is exhibited by untreated smooth surfaces as well. Although this dispersion is only on the order of 0.1%, that is relatively small with respect to the dispersion caused by typical near-surface material variations, it is still higher than the experimental error associated with the measurements themselves, therefore it either has to be corrected for using existing analytical methods or it has to be eliminated through relative measurements which compare the SAW velocities on otherwise identical surface-treated and untreated specimens. The use of laser-interferometric surface wave velocity measurements allows us to observe the small, but very important changes caused by different heat treatments in shot-peened aluminum samples. The presented technique is fully automated and therefore very easy to use. However, the extensive spatial averaging through both lateral and axial scanning, which is necessary to achieve the high accuracy required in such measurements over rough surfaces, is rather time consuming therefore the technique in its present form is limited to laboratory studies. Based on a large number of measurements on surfaces with rms roughness varying between 1 and 10 µm, the relative accuracy of the described laser-interferometric technique is

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approximately 0.01% over a wide frequency range from 1.5 MHz to 15 MHz. The accuracy and relative bandwidth of the present system are slightly better than those of a previously developed laser-ultrasonic system where a pulsed laser was used the excite the SAW and a broadband piezoelectric transducer was used to detect it (Schneider and Schwarz, 1997, Schneider et al., 2000). However, the main advantage of the present system is that these specifications were achieved on fairly rough surfaces, which is a necessary requirement when studying shot-peened and otherwise surface-treated metals. The nominal penetration depth (≈ one wavelength) that can be probed at these frequencies in typical structural materials (c ≈ 3,000 m/s) is between 200 µm and 3 mm. However, it is well known that Rayleigh-type surface waves are mostly sensitive to the top 20% of this nominal penetration range, therefore the actual penetration depth in which the technique can be used to map the elastic properties of the material is somewhere between 40 µm and 600 µm, i.e., it is very well suited to study near surface residual stress and cold work effects in surface-treated metals. By calculating the relative change in velocity, it is possible to better observe the combined effects of surface roughness, residual stress and cold work in the dispersion of SAWs. The experimental results indicate that SAW dispersion in shotpeened aluminum specimens is similar to the theoretically predicted behavior, but the magnitude of the dispersion is larger than expected based on the rms roughness and correlation length of the surface. The causes of the excess dispersion must be due to residual stress and cold work. The depth profile of the SAW velocity distribution can be determined from the measured dispersion spectra by a number of existing theoretical techniques (Hirao et al., 1981, Ditri and Hongerholt, 1996, Delsanto et al., 2000, Glorieux and Gao, 2000). However, such an inversion could not fit within the limits of this paper, the primary purpose of which was to study the variation of SAW dispersion during thermal relaxation as the relative contributions of different dispersion mechanisms gradually change. Furthermore, such an inversion would necessarily neglect the contribution of surface roughness induced scattering to the overall dispersion, therefore it would be somewhat meaningless unless the geometrical artifact is fully corrected for before inversion. We found that the shallow sub-surface region exhibits a higher rate of relaxation and recrystallization than deeper regions. Close to the surface, recrystallization starts at lower temperatures and occurs faster than at larger depths and usually the surface residual stress and cold work are essentially gone after the first heat treatment. This explains why the high-frequency components change faster than the low-frequency components which are affected by the material properties at larger depths. At higher annealing temperatures, the low-frequency dispersion further decreases as a result of continuing partial relaxation and recrystallization deeply below the surface, but at high frequencies the velocity starts to decrease again since the recrystallized grains start to grow in size. Despite the considerable complexity of the problem, the results indicate that ultrasonic evaluation of near surface material variations in shot-peened metals is feasible. Although SAW dispersion is mainly sensitive to cold work effects

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and much less to residual stress effects, it still can be a very useful NDE tool since quantitative assessment of the level and distribution of cold work in surface-treated metals is of primary importance from the point of thermo-mechanical stability of the beneficial residual stresses. Acknowledgment This work was supported by the Air Force Research Laboratory, Metals, Ceramics and NDE Division, under Contract No. F33615-99-C-5803. The authors wish to thank to the Universidad Michoacana de San Nicolás De Hidalgo and CONACYT-MEXICO for their support of Alberto Ruiz during his Ph. D. studies. The authors would also like to acknowledge valuable discussions with Dr. Mark P. Blodgett of AFRL/MLLP. 5. References Auld B. A., Acoustic Fields and Waves in Solids, Malabar, Krieger Publishing, 1990, p. 272302. Carreon H., Nagy P. B., Blodgett M. P., “Thermoelectric Nondestructive Evaluation of Residual Stress in Shot-Peened Metals”, Research in Nondestructive Evaluation, vol. 14, 2002, p. 59-80. Delsanto P. P., Olivero D., Perego G., Scalerandi M., “Acoustoelastic effects in elastic media with nonuniform initial stress”, Research in Nondestructive Evaluation, vol. 12, 2000, p. 105-118. Ditri J. J., Hongerholt D., “Stress distribution determination in isotropic materials via inversion of ultrasonic Rayleigh wave dispersion data”, International Journal of Solids and Structures, vol. 33, 1996, p. 2437-2451. Duquennoy M., Ouaftouh M., Qian M. L., Jenot F., Ourak M., “Ultrasonic characterization of residual stresses in steel rods using a laser line source and piezoelectric transducers”, NDT&E International, vol. 34, 2001, p. 355-362. Eckersley J. S., Champaigne J., Shot Peening: Theory and Application, Gournay-Sur-Marne, IITT-International, 1991. Eguiluz A. G., Maradudin A. A., “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness”, Physics Review B, vol. 28, 1983, p. 728-748. Glorieux C., Gao W., “Surface acoustic wave depth profiling of elastically inhomogeneous materials”, Journal of Applied Physics, vol. 88, 2000, p. 4394-4400. Hirao M., Fukuoka H., Hori K., “Acoustoelastic effect of Rayleigh surface wave in isotropic material”, Journal of Applied Mechanics, vol. 48, 1981, p. 119-124. Kosachev V. V., Shchegrov A. V., “Dispersion and attenuation of surface acoustic waves of various polarizations on a stress free randomly rough surface of solid”, Annals of Physics, vol. 240, 1995, p. 225-265. Krylov V. V., Smirnova Z. A., “Experimental study of the dispersion of a Rayleigh wave on a rough surface”, Soviet Physics Acoustics, vol. 36, 1990, p. 583-585. Lakestani F., Coste J. F., Denis R., “Application of ultrasonic Rayleigh waves to thickness measurement of metallic coatings”, NDT&E International, vol. 28, 1995, p. 171-178. Lavrentyev A. I., Stucky P. A., Veronesi W. A., “Feasibility of ultrasonic and Eddy current methods for measurement of residual stress in shot peened metals”, Review of Progress in Quantitative Nondestructive Evaluation, Melville, AIP, vol. 19, 1999, p. 1621-1628.

Instrumentation Mesure Métrologie. Volume X – noX/2003 Lavrentyev A. I., Veronesi W. A., “Ultrasonic measurement of residual stress in shot peened aluminum alloy”, Review of Progress in Quantitative Nondestructive Evaluation, Melville, AIP, vol. 20, 2001, p. 1472-1479. Lomonosov A., Mayer A. P., Hess P., “Laser controlled surface acoustic waves,” Handbook of Elastic Properties of Solids, Liquids, and Gases, New York, Academic Press, vol. 1, 2001, p. 137-185. Mayer A. P., Lehner M., “Effect of random surface and interface roughness on the propagation of surface acoustic waves”, Waves in Random Media, vol. 4, 1993, p. 321-335. Medik M., Sathish S., “Surface-acoustic-wave studies on single-crystal nickel using Brillouinscattering and scanning acoustic microscope response”, Journal of Applied Physics, vol. 75, 1994, p. 5461-5462. Mutti P., Bottani C. E., Ghislotti G., Beghi M., Briggs G. A. D., Sandercock J. R., “Surface Brillouin Scattering–Extending Surface Wave Measurements to 20 GHz”, Advances in Acoustic Microscopy, New York, Plenum Press, vol.1, 1995, p. 249-300. Nayfeh A. H., Wave Propagation in Layered Anisotropic Media, Amsterdam, North-Holland, 1995.

Pavlakovic B., Lowe M., Disperse 2, London, Imperial College, 2000. Pecorari C., “Modeling variations of Rayleigh wave velocity due to distributions of onedimensional surface-breaking cracks”, Journal of the Acoustical Society of America, vol. 100, 1996, p. 1542-1550. Pecorari C., “Rayleigh wave dispersion due to a distribution of semi-elliptical surface-breaking cracks”, Journal of the Acoustical Society of America, vol. 103, 1998, p. 1383-1387. Pecorari C., “Attenuation and dispersion of Rayleigh waves propagating on a cracked surface: an effective field approach”, Ultrasonics, vol. 38, 2000, p. 754-760. Pecorari C., “Scattering of a Rayleigh wave by a surface-breaking crack with faces in partial contact”, Wave Motion, vol. 33, 2001, p. 259-270. Richardson J. M., “Estimation of surface layer structure from Rayleigh wave dispersion. I. Dense-data case”, Journal of Applied Physics, vol. 48, 1977, p. 498-512. Richardson J. M., Tittman B. R., “Estimation of surface layer structure from Rayleigh wave dispersion. II. Sparse-data case - analytical theory”, Journal of Applied Physics, vol. 48, 1977, p. 5111-5121. Ruiz A. M., Nagy P. B., “Diffraction correction for precision velocity measurements”, Journal of the Acoustical Society of America, vol. 112, 2002, p. 835-843. Schneider D., Schwarz, T., “A photoacoustic method for characterising thin films”, Surface and Coatings Technology, vol. 91, 1997, p. 136-146. Schneider D., Witke, T., Schwarz, T., Schöneich, B., Schultrich, B., “Testing ultra-thin films by laser-acoustics”, Surface and Coatings Technology, vol. 126, 2000, p. 136-141. Tittman B. R., Alers G. A., Thompson R. B., Young R. A., “Characterization of surface anomalies by elastic surface wave dispersion”, 1974 Ultrasonics Symposium Proceedings, IEEE, 1974, p. 561-564. Tittman B. R., Richardson J. M., “Estimation of surface layer structure from Rayleigh wave dispersion. III. Sparse-data case – interpretation of experimental data”, Journal of Applied Physics, vol. 49, 1978, p. 5242-5249. Warren P. D., Pecorari C., Kolosov O. V., Roberts S. G., Briggs G. A. D., “Characterization of surface damage via surface acoustic waves”, Nanotechnology, vol. 7, 1996, p. 295-301. Zhang C., Achenbach J. D., “Dispersion and attenuation of surface waves due to distributed surface-breaking cracks”, Journal of the Acoustical Society of America, vol. 88, 1990, p. 1986-1992.

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