Characterization of rectangular vertical cracks using burst vibrothermographyA. Mendioroz, R. Celorrio, and A. Salazar Citation: Review of Scientific Instruments 86, 064903 (2015); doi: 10.1063/1.4922464 View online: http://dx.doi.org/10.1063/1.4922464 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/86/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in CHARACTERIZATION OF PIEZOELECTRIC STACK ACTUATORS FOR VIBROTHERMOGRAPHY AIP Conf. Proc. 1335, 423 (2011); 10.1063/1.3591883 THE EFFECT OF CRACK CLOSURE ON HEAT GENERATION IN VIBROTHERMOGRAPHY AIP Conf. Proc. 1096, 473 (2009); 10.1063/1.3114289 Optimized deployment of heat-activated surgical staples using thermography Appl. Phys. Lett. 83, 1884 (2003); 10.1063/1.1601305 Scanning thermal microscopy of carbon nanotubes using batch-fabricated probes Appl. Phys. Lett. 77, 4295 (2000); 10.1063/1.1334658 Temperature field visualization in conducting solids using thermographic phosphors Am. J. Phys. 65, 447 (1997); 10.1119/1.18561

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REVIEW OF SCIENTIFIC INSTRUMENTS 86, 064903 (2015)

Characterization of rectangular vertical cracks usingburst vibrothermography

A. Mendioroz,1,a) R. Celorrio,2 and A. Salazar11Departamento de Física Aplicada I, Escuela Técnica Superior de Ingeniería, Universidad del País VascoUPV/EHU, Alameda Urquijo s/n, 48013 Bilbao, Spain2Departamento de Matemática Aplicada, EINA/IUMA, Universidad de Zaragoza, Campus Río Ebro,Edificio Torres Quevedo, 50018 Zaragoza, Spain

(Received 18 March 2015; accepted 1 June 2015; published online 17 June 2015)

We use burst vibrothermography to characterize, i.e., to determine the dimensions and location ofburied vertical cracks of rectangular shape. Surface breaking as well as buried cracks are investi-gated. We calculate the surface temperature distribution generated by a rectangular vertical crackwhen excited by an ultrasound burst of constant power. By fitting synthetic data with added whitenoise, we analyze the effect of the burst duration on the accuracy of the retrieved dimensions anddepth of the crack. We take data on samples containing artificial calibrated vertical cracks. Theresults of the fittings performed on these experimental data show that it is possible to characterizerectangular vertical cracks from burst vibrothermography experiments. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4922464]

I. INTRODUCTIONInfrared thermography (IRT) has arisen as a non-destruc-

tive evaluation technique that is able to detect and eventuallycharacterize shallow subsurface defects. Optically excited IRThas been widely applied to the detection of delaminations,disbonds, or impact damages in aircraft structures.1–3 For suchkind of defects, parallel to the surface, homogeneous illumi-nation of the sample is very efficient in detecting the flaw,since the area of the defect dispersing the thermal energy islarge. On the contrary, in order to detect vertical cracks usingoptically excited thermography, a lateral heat flow with respectto the crack must be established. The crack behaves thermallyas a thermal resistance, whose signature is revealed by anasymmetry in the temperature distribution across the crack.This configuration requires illuminating the sample with afocused spot (or line), either a continuous wave laser spot thatmoves along the sample surface,4–6 a pulsed laser spot,7,8 or amodulated laser spot.9,10 Theoretical studies have shown thatthis kind of technique might be applied to detect buried verticalcracks but the sensitivity decreases strongly with the crackdepth.11 Being the experiments sensitive to the thermal barrierproduced by the crack, these techniques are very well adaptedto the detection of open cracks (strong thermal barrier) and notso sensitive to the presence of kissing cracks (weak thermalbarrier).9 On the contrary, vibrothermography, i.e., ultrasound(US) excited thermography, is a very suited technique to detectkissing cracks. In this technique, the sample is excited withultrasounds and at the defects part of the mechanical energyis converted into thermal energy mainly due to friction be-tween the defect faces. This technique has been used to detectcracks in many kinds of materials, such as metals, polymers,composites, and ceramics,12–19 both in burst13–16,18,19 and lock-in regimes.12,17 Beyond detection, characterizing cracks, i.e.,

a)E-mail: arantza.mendioroz@ehu.es

determining the geometry and location, is highly valuable forindustrial applications. It requires solving the inverse problemthat consists of finding the dimensions and location of thecrack from surface temperature vibrothermography data.

In previous works, we presented a method to size andlocate vertical cracks, based in lock-in vibrothermographyexperiments.20,21 We inverted surface temperature amplitudeand phase data obtained by modulating the amplitude of theultrasounds at nine modulation frequencies. The method onlyassumes prior knowledge of the plane containing the crack,and no particular shape of the heat source is assumed. Thisimplies that the inverse problem is ill-posed and thus the inver-sion algorithm needs to be stabilized. In further investigations,we also showed that buried square cracks can be retrieveddown to depths of twice the side of the square and we founda resolution criterion to obtain separate reconstructions of twoheat sources as a function of their depth.22 The main drawbackof this approach is that data taking in the modulated regimeis time consuming, for instance, in order to obtain an averagenoise in the amplitude image of 0.1 mK using a camera witha typical Noise Equivalent Temperature Difference (NETD)of 25 mK and at frame rate of 100 Hz, the data acquisitiontakes about 40 min. On the contrary, burst vibrothermographyis a fast technique. Burst durations range typically between50 ms and a few tens of seconds, so the total duration of theexperiment takes usually less than one minute. The attempts tocharacterize cracks from burst vibrothermography have beenfocused on finding correlations between the defect heating andits morphology, using finite element modeling.23–25 Analyticalmodeling of the surface temperature evolution arising froma particular crack in a burst experiment has been performedin order to connect the vibrational amplitude and temperaturerise at the surface.26 Approximate analytical models consider-ing the limiting cases of point-, line-, and plane-heat sources(1, 2, and 3-dimensional heat generation) have been used todistinguish between real defects and false positives in burst

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064903-2 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 1. (a) Geometry of a sample containing a vertical buried crack of irregular shape. (b) Cross section of a rectangular vertical crack. The origin of thereference frame is at the surface, centered with the rectangular crack.

experiments.27 The first quantitative comparison between theresults of an analytical model and experimental burst vibroth-ermography data was presented in 2002.28 In that work, the sur-face temperature distribution generated by an inclined crackin the burst regime was calculated together with experimentalresults, showing good agreement with the predictions of themodel.

In this work, we present quantitative characterizationsof rectangular vertical cracks from burst vibrothermographydata. Both surface breaking and buried cracks have been inves-tigated. First, we calculate the evolution of the surface temper-ature distribution corresponding to rectangular vertical cracks.Then, by inverting synthetic data with added white noise, andassuming a rectangular shape of the heat source, we analyzethe influence of the burst duration on the three fitting parame-ters describing the crack: the width, the height, and the depth.This inversion does not need regularization because it onlycontains three unknowns. In order to check experimentally thevalidity of the method, we have performed burst vibrothermog-raphy experiments on samples containing artificial calibratedheat sources. The results of the fittings obtained from exper-imental data confirm it is possible to characterize the crackwith different durations of the burst and opens the door tousing burst vibrothermography to quantitatively characterizevertical cracks.

II. THEORY AND SIMULATIONS

We calculate in this section the evolution of the surfacetemperature distribution produced by rectangular vertical heatsources generating a homogeneous heat flux, I, during a timeinterval τ. They represent rectangular vertical cracks in thepresence of a burst ultrasound excitation of constant intensityand time duration τ.

We start the calculation considering a point-like, instanta-neous heat source, located at r⃗ ′ delivering an energy Q at timet ′ in an infinite medium of thermal diffusivity D, density ρ, andspecific heat c. The temperature at any point r⃗ of the materialat t > t ′ is given by the Green function29

T(r⃗ , t) = Q

8ρc[πD (t − t ′)]3/2 e−|r⃗−r′|2

4D(t−t′) . (1)

Then, we take into account the time duration of the emis-sion: we consider that the point-like heat source delivers en-ergy with constant power P during the time interval [0, τ], τbeing the time duration of the burst. The temperature at anypoint r of the infinite medium of thermal conductivity K canbe written as follows:27,30

T(r⃗ , t) = P4πK |r⃗ − r⃗ ′|Erfc

|r⃗ − r⃗ ′|√

4Dt

0 ≤ t ≤ τ, (2a)

T(r⃗ , t) = P4πK |r⃗ − r⃗ ′|

Erfc

|r⃗ − r⃗ ′|√

4Dt

−Erfc

|r⃗ − r⃗ ′|4D(t − τ)

t > τ. (2b)

Finally, we consider the effect of the spatial extensionof the heat sources in a semi-infinite medium (the sampleoccupies the region z < 0) as sketched in Figure 1(a). We dealwith vertical rectangular heat sources (contained in the planex = 0) of width w, and height h whose upper side is buried adistance d below the sample surface. This geometry is depictedin Figure 1(b), where the origin of the reference frame is at thesurface, centered with the crack.

We assume the heat flux I is homogeneous over the rect-angle, since this is the easiest configuration to be implementedexperimentally. The effect of the sample surface can be ac-counted for by considering the contribution of reflected imagesof the heat sources at the surface. Moreover, in materials thatare opaque to the infrared radiation, only the surface temper-ature is experimentally accessible. In these conditions, thetemperature at the surface can be calculated by integrating thecontribution of point-like heat sources covering the area of thedefect, and multiplying it by a factor 2, which accounts by thepresence of the sample surface,

T(x, y,0, t)

=I

2πK

−d−h−d

b/2−b/2

Erfc√

x2+(y−y′)2+z′2√4Dt

x2 + (y − y ′)2 + z′2

dy ′ dz′

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064903-3 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 2. Simulations of the surface temperature for two surface breaking (d = 0) rectangular heat sources of the same height h = 1 mm and widths: w = 1 mm(solid line) and w = 2 mm (dashed line), calculated for burst durations of τ= 100 ms (blue), τ= 2 s (green), and τ= 20 s (red). In (a) and (b), natural logarithms ofthe normalized temperature calculated at t = τ, along the OX and OY axes, are represented, respectively. (c) Evolution of the normalized temperature calculatedat (0,0,0) as a function of the normalized time, t/τ.

T(x, y,0, t)

=I

2πK

−d−h−d

b/2−b/2

Erfc√

x2+(y−y′)2+z′2√4Dt

x2 + (y − y ′)2 + z′2

dy ′ dz′

−−d

−h−d

b/2−b/2

Erfc√

x2+(y−y′)2+z′2√4D(t−τ)

x2 + (y − y ′)2 + z′2

dy ′ dz′

t > τ. (3b)

Next, we present some simulations to illustrate how vari-ations in the three parameters describing the crack (widthw, height h, and depth d) affect the surface temperature fordifferent burst durations. We show the most relevant anduncorrelated information of the surface temperature evolution,which also exhibits the highest signal to noise ratio: (a) profilesof the natural logarithm of the temperature along the OX andOY axes (perpendicular to the crack through the center andalong the crack direction, respectively), obtained at the endof the burst t = τ and (b) the temperature evolution (timinggraphs) at the origin (0,0,0), right on top of the crack, as afunction of the normalized time, t/τ. All surface temperaturedata are normalized to the temperature value at the origin ofcoordinates (see Figure 1(b)) and at the end of the burst: Tn

=T (x, y,0, t)T (0,0,0,τ) . The material parameters used in the simulations

are those of AISI 304 stainless steel (D = 4 mm2/s and K = 16W/mK), the material our samples are made of. In this section,the standard 1 mm side square crack will be represented by acontinuous line, whereas rectangular cracks will be shown bya dashed line.

In Figure 2, we present simulations for h = 1 mm heightcracks of two different widths (w = 1 and 2 mm) when theyreach the sample surface (d = 0). Three burst durations areconsidered: τ = 100 ms (blue), τ = 2 s (green), and τ = 20 s(red). As expected, the differences are very noticeable alongthe OY axis, but the effect of width can also be noticed in theOX direction, perpendicular to the crack plane: the wider thecrack the wider the profiles. Anyway, the effect along the OXdirection is more pronounced for long burst durations sincethe contribution of the heat sources located far away from theOX axis (the side ends of the crack) does not reach the OXaxis by the end of a short burst. This can also be understoodin terms of the frequency content of the bursts: longer burstsextend the low frequency content of the excitation, makingthe profiles at t = τ more sensitive to the contributions fromdistant sources. The timing graphs in Figure 2(c) show that thenormalized temperature rise and fall at the origin are slower thefarther there are heat sources contributing to the temperatureevolution.

In Figure 3, we analyze the effect of the height of the heatsource. Figures 3(a) and 3(b) show OX and OY profiles forheat sources of the same width w = 1 mm, and two differentheights h = 1 and 2 mm, when they reach the sample surface.

In both spatial profiles, the height of the cracks is barelydistinguishable close to the origin, but further away theybecome wider the taller the heat source. The behavior ofthe timing graphs, shown in Figure 3(c), is equivalent to thecase of different widths (Figure 2(c)): if the heat source islarge, the time evolution of the temperature at the origin isslow.

Finally, in order to show how the surface temperaturechanges for different heights of the heat source when it isburied below the surface, in Figure 4 we present simulations

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064903-4 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 3. The same as in Figure 2 for two surface breaking rectangular heat sources of the same width w = 1 mm and heights h = 1 mm (solid line) and h = 2 mm(dashed line).

corresponding to heat sources of the same width (w = 1 mm)and different heights (h = 1 and 2 mm) when they are buriedd = 1 mm below the surface. As before, burst durations of100 ms, 2 s, and 20 s are considered.

Distinguishing between different heights is the most chal-lenging task, since the deeper side of the heat source is themost inaccessible information. If compared with the results

in Figure 3, the differences between the three geometries areless pronounced for deeper cracks. Moreover, the deeper thecrack, the rounder the profiles close to the origin. In Figure4(c), we show the timing graphs, calculated at the origin (0,0,0)corresponding to the same geometries and burst durations of100 ms, 2 s, and 20 s. A comparison with the simulationsshown in Figure 3(c) indicates that the effect of depth is very

FIG. 4. The same as in Figure 2 for two rectangular heat sources buried at d = 1 mm and having the same width w = 1 mm and heights h = 1 mm (solid line)and h = 2 mm (dashed line).

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064903-5 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

noticeable in the timing graphs; actually, for the deepest crackand the shorter burst duration, the maximum temperature riseoccurs after the burst is over.

All these simulations suggest that any burst durationmight be adequate to retrieve the dimensions of the rectangularcrack. If we focus on the simulations shown in Figure 4,for short burst durations (τ = 100 ms), the profiles and thetiming graphs during the heating period barely differ fromeach other since the information coming from the deepestlocations has not arrived to the sample surface by the end ofthe burst. However, the timing graph during the cooling periodis sensitive to the height of the heat source. At intermediateburst durations (τ = 2 s), both the profiles and the timing graphare sensitive to the heat source height. For long bursts (20 s),the timing graphs become less sensitive since the “effectivedistance” of all the locations in the heat sources is smallcompared to the thermal diffusion length associated to the burstduration Lc =

√τD.31 However, the profiles at the end of the

burst are very sensitive to the dimensions since, by the end ofa long burst, there has been time enough for the informationcoming from the far locations of the heat source to reach thesample surface.

In Sec. III, we check the adequacy of the information wehave presented to retrieve the geometrical parameters of thecrack, depending on the burst duration.

III. FITTINGS OF SYNTHETIC DATA

In order to analyze the effect of the time duration ofthe burst on the retrieved parameters characterizing the crack(width w, height h, and depth d), we have fitted syntheticdata obtained by applying Eq. (3) with added white noise.The fittings have been performed using a regular Levenberg-Marquardt algorithm. The data we enter into the algorithm arethe natural logarithm of the normalized surface temperaturealong the OX and OY profiles calculated at the end of the burst,together with the normalized timing graph at (0,0,0). Spatialprofiles are calculated at positions separated by 100 µm, whichroughly represents the highest spatial resolution attainablewith our IR camera. In order to be conservative, the uniformwhite noise we consider is 5% of the maximum normalized(1 K) temperature rise. The noise in a real experiment is about

100 mK so our noise level corresponds to a maximum signal of2 K, which is representative of a low temperature rise in exper-imental situations with ultrasound powers of about 250 W.The reason why we use natural logarithms of the temperatureprofiles, rather than the temperatures themselves, is related tothe difficulty of retrieving the deepest part of the rectangle.Any experimental technique relying on a diffusive field hasto deal with this limitation. The natural logarithm of the pro-files enhances the differences in the signal corresponding todifferent heights of the defects, which appear in the slopes, farfrom the origin (see Figures 3 and 4), where the temperatureis decreasing and the bare temperature values are all close toeach other and to the noise level.

We start showing the effect of the noise in the temperaturedata. We consider a rectangular heat source of widthw = 1 mmand height h = 2 mm reaching the surface (d = 0) and a burstduration τ = 100 ms. In Figures 5(a) and 5(b), we presentnoise-free Tn profiles (continuous lines) along the OX andOY directions calculated at t = τ, together with noisy datacorresponding to 5% added white noise (dots).

As can be observed, as the distance to the origin increases,the temperature decreases and the noisy signal is dominatedby noise. Fittings performed by taking data up to differentdistances allow us to propose the following general rule ofthumb: we keep data up to a distance/time where the signallevel equals three times the noise level of the data. For instance,in the data shown in Figures 5(a) and 5(b), we would keep dataup to 1 mm and in Figure 5(c) up to 400 ms.

To illustrate the effect of the burst duration, we have fitteddata corresponding to the previously mentioned rectangle(h = 2 mm and w = 1 mm) when it reaches the sample surface(d = 0) and when it is buried d = 1 mm below the surface.Burst durations of 0.1, 0.25, 0.5, 1, 2, 4, 8, 16, and 32 s havebeen considered. For each configuration and burst duration,we have generated 10 different random noise files that weadd to the noise free data. We have fitted the 10 noisy datasets and we have performed a statistical analysis of the fittedparameters. In Figure 6, we show, as an example, simulated(dots) and fitted (continuous lines) data for burst durations of0.5 and 16 s corresponding to the surface breaking rectangle.The corresponding values of the fitted parameters are alsoincluded. As can be observed, the quality of the fittings is good

FIG. 5. Simulations of the natural logarithm of the normalized temperature along the OX (a) and OY (b) axes and (c) normalized temperature evolution at(0,0,0) corresponding to a surface breaking heat source of width w = 1 mm and height h = 2 mm. Noise free data (solid line) and data affected by 5% uniformnoise (symbols).

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064903-6 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 6. Synthetic temperature data affected by 5% white noise (symbols) and fittings to Eq. (3) (solid lines) corresponding to a surface breaking crack of widthw = 1 mm and height h = 2 mm for two burst durations: (a) τ= 0.5 s and (b) τ= 16 s. On top OX profiles, in the middle OY profiles, and at the bottom timinggraphs.

and the retrieved parameters are in very good agreement withthe real values.

Figure 7 shows the results of the average fitted parameters,together with the corresponding relative statistical errors whenthe crack is buried 1 mm below the sample surface. As canbe seen, the relative statistical errors in the estimation of theheight h and the width w of the heat source are around 5%,whereas the corresponding value for the depth is estimatedwith a lower statistical error, about 3%. In the case of a surface-breaking crack of the same dimensions, the relative statisticalerrors are mostly independent of the burst duration with valuesbelow 3% for the height and 1% for the width (not shown inthe figures).

All these results allow us to conclude that, for noise levelslimited to 5%, any burst duration might be adequate to estimatethe geometrical parameters of a rectangular heat source from

the three temperature files described above. However, thisstatement needs some comments:

(a) Regarding the spatial part of the information enteringthe algorithm, the number of data is determined by the spatialresolution of the camera and by the maximum distance ofusable data. As mentioned above, a sensible choice of the dataused is essential for a good estimation of the parameters. Thegeneral rule of thumb is discarding temperature data belowthree times the noise level in the experiment.

(b) Since the sensitivity to the geometrical parameters ofthe crack switches from space to time information along theburst duration, in order to trust any burst for the estimationof the parameters, the number of data in the timing graphmust be of the same order as the number of data in the pro-files. The information entering the fitting algorithm is thenadequately balanced; otherwise, the largest data set (sensitive

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064903-7 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 7. Average values of the retrieved parameters and relative statisticalerrors corresponding to a crack of width w = 1 mm and height h = 2 mm,buried d = 1 mm below the surface as a function of the burst duration.

or insensitive at the applied burst) would dominate the fittingwith no guarantee on the quality of the final fitted parameters.

IV. EXPERIMENTS

In order to test the method experimentally, we took dataon samples containing calibrated rectangular and vertical heat

sources that are activated in the presence of ultrasounds. Thesamples have been described elsewhere.20–22 They consistof two equal AISI 304 stainless steel parts machined witha flat common surface where we place 38 µm thick Curectangular slabs of different dimensions. The two parts areattached by screws. The geometry of the samples is sketchedin Figure 8.

When we launch the ultrasounds, there is friction betweenthe Cu slabs and the steel parts. In order to prevent contactbetween the steel parts themselves, we place two more Cuslabs at the back side of the flat surface so that the heat gener-ated there does not affect the temperature at the measuringsurface. We excite the sample with ultrasounds using a UTvisequipment from EDEVIS at a fixed US frequency (between 15and 25 kHz) that is determined experimentally by performing afrequency sweep at a constant US amplitude. In our samples,the optimum US frequency is about 23 kHz. The US powerranges from 200 to 500 W, depending on the depth of the Cuslab and the duration of the burst. We use an Al film betweenthe sample and the sonotrode to improve mechanical coupling.The thermal energy generated at the Cu films diffuses intothe material and reaches the measuring surface, which wascovered with a thin layer of black paint in order to maximizethe IR emissivity of the sample. This IR radiation was capturedby an IR camera (JADE J550M from Cedip) working in the3.5-5 µm range, with a NETD of 25 mK, provided with a50 mm focal length. Approaching the sample to the minimumworking distance of the lens (about 35 cm), each pixel in thedetector averages the temperature over a 135 µm side squarein the sample surface.

Quantitative analysis of burst vibrothermography data re-quires special care in the data taking process. The assump-tion of constant flux along the heat source relies on applyinga constant ultrasound intensity and on keeping a constantcoupling between the sonotrode and the sample. Keeping thecoupling constant can be difficult to accomplish, especially forlong and/or high intensity bursts, because of eventual relativemotion happening between the sample and the sonotrode. Thelack of a constant coupling has an enormous impact on thetiming graphs and can lead to poor quality fittings and mean-ingless values of the fitted parameters.

FIG. 8. (a) Diagram showing a Cu film acting as a calibrated heat source in the interior of the sample under ultrasound excitation. (b) Diagram of the sonotrodein contact with the sample closed (the two halves joined with screws), as viewed by the camera.

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064903-8 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

FIG. 9. Experimental temperature data (symbols) and fittings to Eq. (3) (solid lines) corresponding to a rectangular Cu slab of width w = 1.2 mm and heighth = 1.8 mm, buried d = 250 µm below the sample surface, for three burst durations: (a) τ= 250 ms, (b) τ= 1 s, and (c) τ= 3 s. On top OX profiles, in the middleOY profiles, and at the bottom timing graphs.

Following this precaution, we recorded data on samplescontaining rectangular 38 µm thick Cu slabs of differentdimensions and located at different depths. As an example,in Figure 9, we show experimental normalized data (dots) andfittings to Eq. (3) (continuous lines) corresponding to a copper

TABLE I. Values of the real dimensions and depth of the Cu slab togetherwith values of the retrieved parameters.

τ (s) h (mm) w (mm) d (µm)

0.25 1.9 1.1 2000.5 1.9 1.5 2351 1.7 0.82 2902 2.1 0.97 2003 1.9 0.98 2404 1.8 0.99 250Real parameters 1.8 1.2 280

slab with h = 1.8 mm, w = 1.2 mm, and d = 250 µm, obtainedfor burst durations of τ = 250 ms (Figure 9(a)), τ = 1 s (Figure9(b)), and τ = 3 s (Figure 9(c)), together with the values of thefitted parameters. The typical maximum temperature rise wasabout 5 K.

Following the guidelines presented at the end of Sec. III,we found sensible values of the fitting parameters, in good

TABLE II. Values of the real dimensions and depth of the Cu slab togetherwith values of the retrieved parameters.

τ (s) h (mm) w (mm) d (µm)

0.5 4.3 1.4 4001 3.5 1.2 4602 3.7 1 480Real parameters 3.5 1.1 450

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064903-9 Mendioroz, Celorrio, and Salazar Rev. Sci. Instrum. 86, 064903 (2015)

agreement with the real dimensions of the copper slabs.Moreover, we buried this Cu slab deeper below the surface(d = 280 µm) and we excited the sample with a large setof burst durations. A summary of the retrieved parametersis given in Table I. We also used a taller slab (w = 1.2 mmand h = 3.5 mm), buried even deeper (d = 450 µm) below thesurface. The obtained parameters are summarized in Table II.As can be observed, the lower part of the heat source can beretrieved down to depths of about 4 mm below the surface fromexperimental data. We believe that these results are promisingregarding the possibility of characterizing vertical cracks fromburst vibrothermography data.

V. SUMMARY AND CONCLUSIONS

We have calculated the evolution of the surface temper-ature distribution of samples containing rectangular verticalcracks when excited by an ultrasound burst of constant inten-sity. The surface temperature simulations obtained for differentdimensions and locations of the crack indicate that, for moder-ate noise levels, it is possible to characterize it for any durationof the burst using OX and OY profiles obtained at the end ofthe burst and the timing graph obtained at the origin of coordi-nates. This is possible because the sensitivity to the heat sourceparameters moves from the timing graph (cooling period) atshort burst durations to the profiles at long burst durations, bothtiming graph and profiles being sensitive at intermediate burstdurations. This hypothesis has been confirmed by analyzingthe values of the fitted parameters obtained from syntheticdata affected by 5% white noise. The fittings performed onexperimental results obtained from samples containing cal-ibrated heat sources are in very good agreement with theparameters describing the real cracks. Although these resultsare promising regarding the characterization of vertical cracksusing burst vibrothermography, the approach presented in thiswork is restricted to a particular geometry and heat flux distri-bution. This limits its applicability to the characterization ofreal cracks, in the present form. This restraint can be over-come if no particular geometry and heat flux distribution ofthe crack are assumed for the fitting. As mentioned in theIntroduction, the inverse problem consisting of finding thegeometry and location of the crack from surface tempera-ture vibrothermography data is ill-posed, and the inversion isunstable. We are currently working on a stabilized inversionalgorithm to retrieve the geometry and location of real verticalcracks.

ACKNOWLEDGMENTS

This work has been supported by the Ministerio de Cien-cia e Innovación (Nos. MAT2011-23811 and MTM2013-40842-P), FEDER, Gobierno Vasco (No. IT619-13), Diput-ación General de Aragón (Grupo consolidado PDIE), andUniversity of the Basque Country (No. UPV/EHU UFI11/55).

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