Int J Thermophys (2015) 36:1208–1216DOI 10.1007/s10765-014-1679-0

Defect Characterization from Lock-inVibrothermography Data

A. Mendioroz · A. Castelo ·R. Celorrio · A. Salazar

Received: 15 November 2013 / Accepted: 28 June 2014 / Published online: 18 July 2014© Springer Science+Business Media New York 2014

Abstract Amethod to characterize vertical cracks which combines lock-in vibrother-mography experiments and a stabilized inversion algorithm is presented.The capabilityof the method to retrieve the shape and location of square cracks buried at increasingdepths is analyzed by inverting synthetic data with added white noise. Moreover, theability of the method to resolve two square buried cracks as a function of depth isstudied. A relationship between the depth of the cracks and the minimum distancebetween them in order to obtain separate reconstructions is found. The predictions ofthe inversions with synthetic data have been checked by inverting experimental dataobtained from steel samples containing calibrated cracks. In order to prevent the sam-ple from being damaged and to reach the steady state faster, low ultrasound excitationpowers are used. The signal-to-noise ratio corresponding to the resulting weak signalis improved by processing a large number of infrared images using the lock-inmethod.The experimental results confirm that the method is able to reconstruct square heatsources whose depth is twice their lateral dimension, and that the resolution conditionholds.

Keywords Infrared thermography · Nondestructive evaluation ·Ultrasonic thermography · Vibrothermography

A. Mendioroz (B) · A. Castelo · A. SalazarDepartamento de Física Aplicada I, Escuela Técnica Superior de Ingeniería,Universidad del País Vasco UPV/EHU, Alameda Urquijo s/n, Bilbao 48013, Spaine-mail: arantza.mendioroz@ehu.es

R. CelorrioDepartamento de Matemática Aplicada, EINA/IUMA, Universidad de Zaragoza,Campus Río Ebro, Edificio Torres Quevedo, Zaragoza 50018, Spain

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1 Introduction

Vibrothermography is a nondestructive evaluation test inwhich the sample ismechani-cally excited bymeans of ultrasound. In the presenceof defects such as cracks or delam-inations, mechanical energy transforms into thermal energy due to friction betweenthe faces and/or plastic deformation in the surrounding area. This thermal energydiffuses into the sample and produces a surface temperature distribution that can bemeasured using an infrared (IR) camera. This technique has demonstrated its ability todetect defects in different materials, such as metals, ceramics, carbon fiber reinforcedplastics, and honeycomb core sandwich structures [1–4].

In the last decade, correlations between defect heating and morphology have beensought usingfinite element simulations [5,6].However, the relationship betweendefectmorphology and location and the measured surface temperature distribution has beenbarely analyzed [7]. In our approach, we assume heat is generated at the defectsand study the surface temperature distribution it produces in lock-in vibrothermogra-phy experiments. In previous works, we modeled the surface temperature distributioncorresponding to vertical buried heat sources (defects) [8] and developed an inversionalgorithm to retrieve the shape and location of the heat source [9]. In this work, we ana-lyze the capability of the method to retrieve vertical square heat sources at increasingdepths by performing inversionswith synthetic datawith addedwhite noise.Moreover,we study its ability to resolve two square heat sources buried beneath the surface. Thesepredictions have been confirmed by the reconstructions obtained from experimentaldata corresponding to samples containing calibrated heat sources.

2 Direct and Inverse Problem

2.1 Direct Problem

We calculate in this section the surface temperature distribution corresponding to tworectangular modulated heat sources, contained in plane �(x = 0), perpendicular tothe sample surface (z = 0), of width wi , height hi , whose centers are located aty-coordinates li , and submerged distances di from the sample surface, covering atotal area Ω = ∑2

i=1 Ωi . These heat sources are intensity modulated at frequency fand represent the behavior of closed cracks in the presence of amplitude-modulatedultrasound. The sample is thick enough to be considered as semi-infinite. Assumingadiabatic boundary conditions and homogeneous heat sources, the surface temperaturedistribution can be represented in a complex form as [8]

T (x, y, 0) =∫ ∫

Ω

Q

4πK

e−q√

(�r−�r0)2√

(�r − �r0)2dS0

=2∑

i=1

li+wi /2∫

li−wi /2

−di∫

−(hi+di )

Q

4πK

e−q√x2+(y−y0)2+z20

√x2 + (y − y0)2 + z20

dy0dz0, (1)

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where Q is the maximum emitted power density,K is the thermal conductivity, andq = √

2π i f/D is the thermal wave vector, with D the thermal diffusivity and i theimaginary unit. Both the amplitude and phase of the oscillating surface temperature canbe computed from Eq. 1. Moreover, �r = (x, y, 0) is the position vector at the samplesurface and vector �r0 = (0, y0, z0) represents the locations of the heat sources.

2.2 Inverse Problem

We now consider the inverse problem consisting of retrieving the geometry and inten-sity of heat sources giving rise to a certain temperature distribution in a lock-in vibroth-ermography experiment. We thus look for a solution of integral Eq. 1. In order tolinearize the problem, we introduce a function describing position and frequency-dependent heat sources, Q fk (�r0), and we integrate over the whole � plane,

Q fk (�r0) = Ω(�r0) I fk . (2)

Here Ω(�r0) is a position-dependent characteristic function, common for all mod-ulation frequencies, whose value is 1 within area Ω and zero elsewhere, and I fk is afrequency-dependent intensity, allowing the intensity of the ultrasound being differentat different modulation frequencies.

Now, we deal with the fact that available data are noisy. In the presence of noise,δ, the first equality in Eq. 1 does not hold anymore, it becomes an approximation,to be understood as a minimization problem. Moreover, since we are dealing withan ill-posed problem, the solution is extremely sensitive to noise in the data and theinversion is unstable. We have stabilized the inversion by introducing a bounded valueregularization functional, total variation, (TV(g) = ∫ ∫

π|∇g| dS) [10], multiplied by

a regularization parameter, α, which determines the size of the regularization termwith respect to the residual to be minimized. The introduction of a regularization termstabilizes the inversion but also introduces an error in the solution, so that the retrievedintensities, I δ,α

fk, and domains, Ωδ,α , will be affected by noise, δ, and by the value of

α. The function to be minimized, R2α , is then

R2α = αTV(Ωδ,α) +

kmax∑

k=1

∥∥∥T calc

fk (I δ,αfk

Ωδ,α) − T δfk

∥∥∥2. (3)

We have used a domain decomposition algorithm (a nonlinear Gauss–Seidel typemethod) in which we retrieve the intensities and characteristic functions separately, insuccessive iterations. For details concerning the inversion procedure, see Ref. [9].

3 Inversion of Synthetic Data

3.1 Single Square Crack

The aim of this study was to determine the maximum depth of the heat source forwhich a meaningful reconstruction can be obtained. As a standard, we have chosen

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Fig. 1 Reconstructions obtained from synthetic data with 5% (a) addedwhite noise for a square heat sourceburied at different depths. Real squares are represented by a red contour line while the reconstructions aredepicted on a black and white scale, white showing the maximum (1) value of the indicator function andblack the absence of heat sources

a square heat source, 1 mm on a side, located at different depths from the surfaceof an AISI 304 stainless-steel sample (D = 4 mm2·s−1, K = 15W·m−1·K−1), thematerial of which our samples are made. As in the experiments, synthetic data aregenerated at modulation frequencies of (0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, and12.8) Hz, corresponding to thermal diffusion lengths ranging from 0.3 mm to 5 mm.The conclusions of this study will be scalable to squares of different sizes, L , anddifferent sample properties, provided the data used for the inversions are generated atmodulation frequencies corresponding to thermal diffusion lengths representing thesame fractions of the square size from 0.3L to 5L .

First, we generate synthetic data using Eq. 1 with N = 1, h1 = w1 = 1mm,and l1 = 0, and d1 = (0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.6, 2, 2.6, and 3.2) mm. Data arecalculated at positions of the surface separated by 135μm, which is the side of thesquare in the samplewhere each pixel of our infrared camera averages the temperature.Then we add 5 % random white noise (which roughly represents the noise in ourexperiments), and we distribute it homogeneously over the whole frequency range.This procedure gives a larger relative noise at higher modulation frequencies, as inreal experimental data. We finally normalize amplitudes as a way to standardize datainput in the algorithm.

Using the inversion algorithm described in the previous section, we have obtainedthe inversions shown in Fig. 1, which summarizes the scaled results for a square ofside L . The values of the characteristic function, Ωδ,α , are represented in a black andwhite diagram: white corresponding to the maximum value (1) of the characteristicfunction and black (0) the absence of heat sources. The real geometry of the heatsource is shown as a red square contour. As can be observed, the reconstructions arevery good for square depths down to 0.6L . From then on, as the depth increases, thereconstructions become larger than the real squares. The reason for this is that theinformation carried by high frequencies is lost progressively as the depth increases, sodeeper heat sources are reconstructed with a gradually narrower set of low frequencyinformation, which in turn gives wide, poorly defined reconstructions. We set themaximum depth to get a good reconstruction as twice the side of the square.

3.2 Double Square Crack

Next we analyze the spatial resolution for two buried cracks. For this purpose,we pick two equally buried square heat sources, 1 mm on a side, and perform

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Fig. 2 Reconstructions of two squares, 1 mm side, buried 1.6 mm beneath the sample surface and withcenters separated by (a) 2.8 mm and (b) 3.4 mm. Inversions performed from synthetic data with 5 % addedwhite noise. Real heat sources are depicted in a red line contour. (c) Limiting distances between squaresof size L for heat source resolution. Dark gray (light gray) areas represent the locations of heat sourcesresolvable (non-resolvable) with 5 % noise

inversions by changing the distance between them. The inversions have been car-ried out at different depths of the two heat sources, namely, (0.2, 0.6, 1.2, 1.6, 2,2.6, and 3.2) mm, all with 5 % added white noise. As an example of what thesereconstructions look like, in Fig. 2a, b we show the reconstructions correspond-ing to a depth d = 1.6 mm, with the centers of the squares separated by 2.8mm and 3.4 mm, respectively. The real heat sources are depicted by a red contourline.

We have established a resolution criterion by plotting the value of the normalizedintensity, Ωδ,α , between the two maxima as a function of the distance between cen-ters and we have selected a value of 0.3 as the limiting criterion: when the intensityat the center is below this value, the heat sources are resolved. For the crack sep-aration corresponding to Fig 2a, the intensities between maxima are 0.68 and 0.03in Fig. 2b, corresponding to non-resolved and resolved heat sources, respectively.We have followed this procedure for all the depths between 0.2 mm and 3.2 mmlisted above, and have plotted the limiting distance between centers of scaled squaresof size L from data affected by 5 % white noise. The result is depicted in Fig. 2c,where we show the positions of the centers (black lines), together with the contoursrepresenting the limiting situations. Two squares buried the same depth should belocated in the outer darker regions in order to be resolved with this method. Asexpected, the minimum distance between squares to have them resolved increaseswith depth.

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Fig. 3 Experimental thermograms corresponding to a 1 mm side square Cu slab buried 0.1 mm below thesurface, obtained at a modulation frequency or 1.6 Hz: (a) natural logarithm of the normalized amplitudeand (b) phase

4 Experiments

In order to test the inversion algorithmwith experimental data and check the predictionsof inversions performed with synthetic data, we have prepared a sample containingcalibrated heat sources. The sample is composed of two AISI-304 stainless-steel partsjoined by screws with a common flat surface, in which one or two square Cu slabs(38µm thick) are placed representing the defects. When the ultrasound is launchedinto the sample, friction occurs between the Cu slabs and steel flat surfaces. In order toguarantee that friction only takes place at the location of the slabs, two more films areintroduced at the back side, far enough from the measuring surface so that they do notaffect the data coming from the calibrated heat sources at the front side. Ultrasoundexcitation with amplitude modulation at (0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, and12.8) Hz has been performed with UTvis equipment from Edevis. The IR radiation iscaptured by an IR camera (JADE J550M from Cedip) working in the 3.5µm to 5µmrange.

In order to prevent the sample from being damaged [11,12] and to reduce the timeneeded for the sample to reach equilibrium [8], experiments have been carried outwith a low excitation power (below 40 W) instead of the usual powers, above 250 W[8]. The price to be paid is a dramatic reduction of the amplitude of the oscillatingcomponent. In order to maintain a good enough signal-to-noise ratio, we have takenadvantage of the lock-in ability to reduce the noise level by increasing the number ofimages analyzed, Nimages. The average amplitude noise level, 〈Tnoise〉, can be reducedbelow the noise equivalent temperature difference (NETD) of the camera by [13]

〈Tnoise〉 = 2√Nimages

NETD, (4)

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Fig. 4 Real heat sources (red line contour) and reconstructions from experimental data

Fig. 5 Real contours and retrieved heat sources obtained from experimental data corresponding to twosquare 1 mm side Cu slabs, both buried (a and b) 0.2 mm and (c and d) 1.6 mm at different separationsbetween centers: (a) 0.15 mm, (b) 1 mm, (c) 2.5 mm, and (d) 5 mm. Real heat sources are depicted in a redline contour

Accordingly, we have performed experiments by acquiring images at 350 frames/s,during different time intervals depending on the modulation frequency. Moreover, allexperimental phase data were corrected by the frequency response of the equipment.

In order to check the maximum depth for reconstructing square cracks, we haveprepared samples containing a 1 mm side square Cu slab, buried at (0.2, 0.6, 1, 1.6, 2,and 3) mm, in the set of frequencies mentioned previously. As an example, in Fig. 3we show experimental thermograms of the phase and the natural logarithm of thenormalized amplitude obtained at a modulation frequency of 1.6 Hz in the samplecontaining a 1 mm side Cu slab buried 0.1 mm below the surface. The reconstructionsobtained for all the cases studied are summarized in Fig. 4. As can be observed, thetrend is similar to the inversions performed with synthetic data: down to 0.6 mm,the reconstructions are quite accurate, but as the heat source is buried deeper insidethe material, the reconstructions spreads in an area larger than the actual heat source.On the other hand, the depth of the heat sources is quite well reproduced. We wantto emphasize that we get good reconstructions of heat sources as deep as 2L , fromexperimental data.

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We now check experimentally the results presented in the previous section on thespatial resolution criterion of heat sources at increasing depths (Fig. 2c). To do so,we have picked two depths (0.2 mm and 1.6 mm) and have performed experimentswith two 1 mm on a side heat sources and two separations between centers for eachdepth. One of the separations corresponds to both squares in the “forbidden” light graycentral region (1.15 mm for the shallower and 2.5 mm for the deeper) and the otherone with squares in the “allowed” dark sides (2 mm for the shallower and 5 mm forthe deeper). The results are depicted in Fig. 5. On the left, we show the cases in whichno resolution is expected, and on the right, the separations for which we expect theheat sources to be resolved.

The intensities between maxima in the four figures are 0.99, 0.25, 0.85, and 0.24,respectively: above and below the limiting value of 0.3 as expected in each case. Theseresults confirm experimentally the prediction on the allowed/forbidden regions for twosquares to be resolved with 5 % noise.

5 Summary and Conclusions

We have analyzed the ability of lock-in vibrothermography combined with a stabilizedinversion algorithm to retrieve buried square heat sources as a function of depth. Theanalysis performed with synthetic data (with added white noise) indicates that theaccuracy of the reconstruction of deep cracks is limited by the available informationat high frequency that is reduced as the heat source is buried deeper below the surface.The maximum estimated depth of a square heat source of side L to be retrieved bythe method from data affected by 5 % white noise is 2L . We have also determined theallowed and forbidden distances between two equally buried square heat sources forthem to be resolved as a function of depth, by inverting temperature data with 5 %white noise. The predictions of inversions from synthetic data have been validated byinverting experimental data obtained from samples containing calibrated heat sources.Experiments have been performed at low intensity excitation in order to facilitatedata taking at a steady state and also to reduce sample aggression. The inversions ofexperimental data confirm the predictions of the inversions from synthetic data andcertify that the method is able to reconstruct square heat sources as deep as twice theside of the square from experimental data. The results are very promising regardingthe possibility of characterizing real vertical defects from lock-in vibrothermographydata.

Acknowledgments This work has been supported by theMinisterio de Ciencia e Innovación (MAT2011-23811 and MTM2010-16917), by Gobierno Vasco (IT619-13), and by Diputación General de Aragón.

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