J Nondestruct Eval (2016) 35:22 DOI 10.1007/s10921-016-0344-x

Fast Characterization of the Width of Vertical Cracks UsingPulsed Laser Spot Infrared Thermography

N. W. Pech-May1,2 · A. Oleaga1 · A. Mendioroz1 · A. Salazar1

Received: 30 September 2015 / Accepted: 9 March 2016© Springer Science+Business Media New York 2016

Abstract In-service non-destructive detection of cracks isa challenging task for industries to prevent failures. In thelast decades several methods based on infrared thermogra-phy have been proposed to detect vertical cracks. In a recentpaper, the authors used a lock-in thermography setup withfocused laser excitation to characterize the width of infinitevertical cracks accurately. As this method is very time con-suming, we propose in this work to measure the width ofan infinite vertical crack using pulsed laser spot infraredthermography. A semi-analytical solution for the surfacetemperature of a sample containing such a crack when thesurface is illuminated by a pulsed Gaussian laser spot closeto the crack is obtained. Measurements of the surface tem-perature on samples containing calibrated cracks have beenperformed using an infrared camera. A least square fit ofthe surface temperature is used to retrieve the thickness ofthe crack. Very good agreement between the nominal andretrieved thickness of fissure is found, even for widths downto 1 μm, confirming the validity of the model.

Keywords Pulsed infrared thermography · Crackdetection · Crack sizing · Nondestructive evaluation

B A. Salazaragustin.salazar@ehu.es

1 Departamento de Física Aplicada I, Escuela de Ingeniería deBilbao, Universidad del País Vasco UPV/EHU, AlamedaUrquijo s/n, 48013 Bilbao, Spain

2 Department of Applied Physics, CINVESTAV UnidadMérida, carretera Antigua a Progreso km6, A.P. 73Cordemex, 97310 Mérida, Yucatán, Mexico

1 Introduction

The growing necessity of in-service non-destructive testingand evaluation of surface breaking cracks in a wide variety ofdevices has been a challenging task for modern industries andlaboratories. Several well established methodologies like dyepenetrant, magnetic particles, eddy currents, and x-ray havebeen proposed and used to detect fissures as well as defectsin materials. Alternative techniques like vibrothermographyor optically stimulated thermography have gained attentionin the recent decades due to their performance on crackdetection and characterization [1–4]. In vibrothermography,the sample is excited by ultrasonic waves and the contact-ing surfaces of mechanical discontinuities produce heat thatpropagates to the sample surface indicating the presence ofthe defect on a cold background. This technique has receivedan increasing attention in the last decade because of its abilityto detect and characterize cracks in a wide variety of materialscovering metals, ceramics, polymers and composites [3–9].However, it might be difficult for this technique to detectopen cracks with barely rubbing surfaces. On the other hand,it needs the ultrasonic transducer to be in contact with thesample surface, thus reducing its versatility. Optically stim-ulated infrared thermography is fully noncontact instead. Inthis case, the presence of the defect produces just a pertur-bation of the existing surface temperature field generated bythe external optical excitation. Moreover, the spatial config-uration of the illumination strongly affects the detectabilityof cracks. For example, considering vertical cracks, if thesample is excited by a homogeneous illumination which pro-duces a heat flux perpendicular to the sample surface, thecrack will barely disperse this flux, thus a negligible signa-ture will be present on the surface temperature distribution,making almost impossible to detect vertical cracks. This kindof cracks can be detected only when an asymmetry in the

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heat flux is produced. This idea was exploited in the ninetieswhen the so-called flying spot method was introduced [10].It consists of heating the sample with a moving laser spotor line and detecting the time evolution of the surface tem-perature with an infrared camera [2,10–17]. The excitationcan be a continuous wave [10–14], modulated [16] or pulsedlaser illumination [2,15–17]. This method allowed detectingcracks with openings of a few micrometers [15].

Once the crack has been detected the interest focuses onthe characterization of its geometrical parameters (depth,length, width, orientation…). In the last years, severalapproaches to crack characterization have been proposed[18–20]. They take advantage of the asymmetry of the tem-perature field at both sides of the crack arising from thethermal resistance produced by the crack, which partiallyblocks heat flux when the laser spot is focused close to thecrack.

In this direction, in a recent work, the authors dealt withthe characterization of infinite vertical cracks using lock-inthermography, which is able to provide surface temperatureamplitude and phase images with a very low noise level [21].We found a semi-analytical expression for the surface tem-perature of a sample containing such a crack when the surfaceis illuminated by a modulated and focused laser spot close tothe crack. By fitting the surface temperature amplitude andphase to the model, the thermal contact resistance Rth of thecrack, which quantifies the width of the crack, was obtained.This method is very accurate, but the data acquisition is verytime consuming (several minutes). That is the reason why wepropose working in the time domain, instead of in the fre-quency domain, using pulsed laser spot thermography to sizethe width of vertical cracks in a fast manner: the data acqui-sition radically drops from several minutes to a few seconds.First, we have found a semi-analytical expression for the timeevolution of the surface temperature of a sample containingsuch a crack when the surface is illuminated by a Dirac-likepulsed and focused Gaussian laser spot close to the crack.The presence of the defect produces an abrupt jump in thetemperature profile at the crack position. The influence of theexperimental parameters (laser beam radius, distance spot-crack, time of measurement and width of the crack) on thejump height is analyzed.

In order to prepare calibrated infinite vertical cracks, verythin metallic tapes down to 1 μm thick are inserted betweentwo identical blocks under pressure. A pulsed laser beam isfocused close to the crack. An infrared video camera recordsthe surface temperature around the crack. A microscope lenswith a spatial resolution of 31 μm is used to collect theinfrared emission from the sample. A least square fit of thetemperature profile crossing the center of the laser spot andperpendicular to the crack is used to retrieve Rth . Very goodagreement between the thickness of the metallic tapes and theobtained Rth is found, confirming the validity of the model.

2 Theory

The diagram shown in Fig. 1 represents the problem ofa semi-infinite and opaque material containing an infinitevertical crack placed at plane y= 0. The sample surface isilluminated by an infinitely brief (Dirac-like) laser pulse ofGaussian profile and energy per pulse Qo. The center of thelaser spot is located at a distance d from the crack and itsradius is a (at 1/e2 of the intensity). Adiabatic boundary con-ditions at the sample surface have been assumed. Under theseconditions, the temperature at any point of the material hasbeen already solved in the frequency domain (see Eq. (9) inRef. [21]), i.e., when the sample surface is heated up by amodulated laser beam of power Po and frequency f (ω =2π f ):

T (x, y, z, ω)

= Po2πK

∫ ∞

0δ J0(δro)

e−(δa)2/8

βdδ

+ sign(y)PoK Rth

π2a2K

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

0dxodyodδe

− 2[x2o+(yo−d)2

]

a2

δ J0(δr1)sign(yo)e−β(|yo|+|y|)

2 + K Rthβ, (1)

where ro = √x2 + (y − d)2 + z2, r1 = √

(x − xo)2 + z2,β = √

δ2 + iω/α, J0 is the Bessel function of zero order,the sign function is equal to +1 for all positive values of itsargument and is equal to -1 for the negative ones, K and α

are the thermal conductivity and diffusivity of the materialrespectively. Finally, Rth is the thermal contact resistanceof the crack related to the air gap thickness L through theequation Rth = L/Kair [22].

Notice that in Eq. (1) there are two summation terms:the first one corresponds to the temperature field on a semi-infinite material without any crack, whereas the second one

y

z

x

Gaussian beam

d

a

crack

Fig. 1 Scheme of a semi-infinite sample which contains an infinitevertical crack (in gray) and that is illuminated by a circular Gaussianbeam

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represents the contribution of the vertical crack to the tem-perature field in the material. This phenomenon can beunderstood in terms of the thermal waves that are reflectedand/or transmitted through the vertical crack, as extensivelyexplained in [21].

It is worth noting that the Laplace transform of the sam-ple temperature in the time domain after a Dirac-like pulseT̄ (x, y, z, s) is obtained from the modulated solution bychanging iω (that appears implicitly in β) by s and Po byQo, the energy of the laser pulse.

T̄ (x, y, z, s)

= Qo

2πK

∫ ∞

0δ J0(δro)

e−(δa)2/8

β1dδ

+ sign(y)QoK Rth

π2a2K

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

0dxodyodδe

− 2[x2o+(yo−d)2

]

a2

δ J0(δr1)sign(yo)e−β1(|yo|+|y|)

2 + K Rthβ1, (2)

where β1 = √δ2 + s/α.

We focus on the Laplace transform of the temperatureprofile along the y-axis

T̄ (0, y, 0, s)

= Qo

2πK

∫ ∞

0δ J0(δ |y − d|)e

−(δa)2/8

β1dδ

+ sign(y)QoK Rth

π2a2K

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

0dxodyodδe

− 2[x2o+(yo−d)2

]

a2

δ J0(δ |xo|)sign(yo)e−β1(|yo|+|y|)

2 + K Rthβ1, (3)

that can be written as T̄ (0, y, 0, s) = T̄1(0, y, 0, s) +T̄2(0, y, 0, s), separating the contribution of the semi-infinitematerial without crack from the contribution of the crack.

The surface temperature evolution in the absence of anycrack is obtained through the inverse Laplace transform ofT̄1(0, y, 0, s), which is the first term in Eq. (3). This isachieved by inverting 1/β1 and solving the resulting integralanalytically [23]

T1(0, y, 0, t) = Qo

2e√

π3t

∫ ∞

0δ J0(δ |y − d|)e−δ2

(a2+8αt

8

)

dδ = 2Qo

e√

π3t

e− 2(y−d)2

a2+8αt

a2 + 8αt, (4)

where e = K/√

α stands for the thermal effusivity of thematerial. It is worth mentioning that this result agrees withthe limiting case (semi-infinite material) of the series solutionobtained by Cernuschi and coworkers for a material illumi-

nated by a Dirac-like laser pulse of Gaussian profile (see Eq.(5) in Ref. [23]).

The inverse Laplace transform of T̄2(0, y, 0, s), whichis the contribution of the vertical crack to the temperatureprofile in the time domain, consists of inverting the factorcontaining β1, which writes

F̄(s) = e−u√

δ2+s/α

2 + K Rth

√δ2 + s/α

, (5)

that is performed by using the partial fractions decompositionand the properties of the Laplace transform [24]

F(t) =√

α

K Rth

e−αtδ2−u2/(4αt)

√π t

− 2α

(K Rth)2 e−αtδ2−u2/(4αt)

e

( √4αt

K Rth+ u√

4αt

)2

erfc

(√4αt

K Rth+ u√

4αt

), (6)

where u = |y| + |yo| and erfc is the complementary errorfunction [25].

On the other hand, the integrals over xo and over δ in thesecond term of Eq. (3) have analytical solutions

Ixo =∫ ∞

−∞dxoe

−2x2o/a2

J0(δ |xo|)

= a√

π√2e−a2δ2/16 I0

(a2δ2

16

), (7)

Iδ =∫ ∞

0dδe−a2δ2/16e−δ2αtδ I0

(a2δ2

16

)

=√

2√αt

1√a2 + 8αt

, (8)

being I0 the modified Bessel function of zero order.Using the results obtained in Eqs. (6)–(8), we can write

an expression for the contribution of the vertical crack to thetemperature profile along the y-axis

T2(0, y, 0, t)

= sign(y)Qo

eπ2at√

α√a2 + 8αt∫ ∞

−∞dyosign(yo)e

− 2(yo−d)2

a2 − u24αt

·[

1 −√

4παt

K Rthe(

√4αt

K Rth+ u√

4αt)2

erfc

(√4αt

K Rth+ u√

4αt

)].

(9)

Finally, the surface temperature along the y-profile in thetime domain is obtained by adding Eqs. (4) and (9)

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T (0, y, 0, t) = 2Qo

e√

π3t

e− 2(y−d)2

a2+2μ2

a2 + 2μ2

+sign(y)Qo

eπ2a√tμ

√a2 + 2μ2

×∫ ∞−∞

dyosign(yo)e− 2(yo−d)2

a2 − u2

μ2

·[

1 −√

πμ

K Rthe

(μ

K Rth+ u

μ

)2

erfc

(μ

K Rth+ u

μ

)],

(10)

where μ = √4αt is the so called thermal diffusion length

[26]. Note that the thermal resistance is correlated to the ther-mal conductivity through the factorK Rth . This means thatnarrow cracks are better detected in high thermal conduct-ing materials (metals, alloys, ceramics…) than in thermalinsulators (polymers, composites…). Besides, the thermaldiffusivity is correlated to the time after the laser pulsethrough the factor αt . Accordingly, under the same exper-imental conditions, for low thermal diffusivity samples theeffect of the crack arises later than for good thermal diffusors.

3 Numerical Simulations

We have performed numerical simulations of the surface tem-perature profile along the y-axis of the semi-infinite cracked

sample, using Eq. (10), to illustrate the effect of the thermalresistance as well as the influence ofd,a and t on the visibilityof the vertical crack. All the simulations are performed con-sidering AISI-304 stainless steel samples (α = 4.0 mm2 s−1

and K = 16 Wm−1 K−1) and Dirac-like laser pulse illumi-nation. It is important to mention that for very small cracks(K Rth < 10−4m), the numerical evaluation of the integralterm in Eq. (10) is more stable when using an asymptoticexpansion of ez

2erfc(z) [27].

In Fig. 2 we show the temperature rise, above the ambi-ent, along the y-axis for an AISI-304 sample containing aninfinite vertical crack at y = 0. The laser spot is centered atd = 1.0 mm with radius a = 0.75 mm. We show the effect ofthe value of the thermal resistance Rth , ranging from 10−3

to 10−6 m2W−1K, at four different times after the heatingpulse.

Notice that there is a discontinuity at the crack position.The larger the thermal resistance, the larger the temperaturejump at any time. However, the normalized temperature con-trast, i.e., the ratio of the temperature jump and the maximumtemperature value, is low for early times (see Fig. 2a) andgrows for medium times (see Figs. 2b and c) to a maximumvalue, then starts decreasing for longer times (see Fig. 2d).Moreover, for long times the temperature rise is too low andhence it will be affected by noise in real experiments.

Fig. 2 Numerical simulationsof the temperature profiles alongthe y-axis for an AISI-304sample containing an infinitevertical crack (y = 0). Thesample is illuminated atd = 1.0 mm by a Dirac-likepulse of Gaussian profile witha = 0.75 mm. We analyze theeffect of the thermal resistanceRth (m2W−1K) at four times: a10 ms, b 70 ms, c 100 ms and d1 s after the heating pulse (Colorfigure online)

0

5

10

15

T (

K)

y (mm)

Rth

10-3

10-4

10-5

10-6

(a)

10 ms

0

0.5

1

1.5

2

T (

K)

y (mm)

Rth

10-3

10-4

10-5

10-6

(b)

70 ms

0

0.5

1

1.5

T (

K)

y (mm)

Rth

10-3

10-4

10-5

10-6

(c)

100 ms

0

0.02

0.04

0.06

0.08

-1 0 1 2 3 -2 -1 0 1 2 3 4

-2 -1 0 1 2 3 4 -6 -4 -2 0 2 4 6 8

T (

K)

y (mm)

Rth

10-3

10-4

10-5

10-6

(d)

1 s

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J Nondestruct Eval (2016) 35:22 Page 5 of 10 22

0

0.2

0.4

0.6

0.8

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

T

Rth

(m2W-1K)

1 s

0.01 s

0.1 s

Fig. 3 Numerical simulation of the dependence of �T on the thermalcontact resistance Rth . Calculations are performed for a cracked AISI-304 sample with d = μ and a = μ/2. Three times after the laser pulseare evaluated (Color figure online)

The normalized temperature contrast at the crack position�T is useful to quantify the strength of the temperature jumpat the crack position and is defined as

�T (t) = T (0, 0+, 0, t) − T (0, 0−, 0, t)

T (0, d, 0, t). (11)

According to Eq. (10), the relative temperature contrast �T

depends on the factor K Rth , on the distance d and on theradius of the laser beam a. However, �T does not depend onthe energy Qo delivered by the laser pulse.

Figure 3 shows the numerical simulations of �T as a func-tion of the thermal resistance performed on AISI-304 with d= μ and a = μ/2. Three times after the laser pulse are con-sidered: 10 ms (green curve), 100 ms (red curve) and 1.0s (black curve). According to the definition of μ = √

4αt ,in the simulations of Fig. 3, early times means using a smalllaser spot impinging the sample very close to the crack, whilelong times means using a large laser spot far away from thecrack. As can be seen, the temperature contrast �T exhibits asigmoid shape. For low thermal resistances Rth (very narrowcracks) there is no temperature contrast and the vertical crackremains undetected. For large values of Rth (thick cracks), thetemperature contrast saturates (�T → 0.8), indicating thatalthough these large cracks are easy to detect, it is difficultto quantify Rth . In fact, the highest sensitivity to the thermalresistance appears for intermediate Rth values. Nevertheless,notice that increasing the measurement time by two ordersof magnitude, the sensitivity to Rth shifts to higher values byone order of magnitude. This result means that using earlytimes after the heating pulse (together with the laser spottightly focused near the crack) is better to detect and char-acterize narrow cracks. In the following, we will explore theeffect of the other parameters involved in �T in order to beable to state guidelines for the characterization of verticalcracks.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

T

t (s)

10-5

10-4

10-3

Fig. 4 Numerical simulation of the dependence of �T on the mea-surement time t . We have considered a cracked AISI-304 sample withd = 1.0 mm and a = 0.75 mm. Three different thermal resistances: 10−5,10−4 and 10−3 m2W−1K are studied (Color figure online)

0

0.2

0.4

0.6

0.8

0 1 2 3 4

T

a ( )

10-3

10-4

10-5

Fig. 5 Numerical simulations of �T as a function of the radius ofthe spot. We have considered a cracked AISI-304 sample with d =μ = 1.26 mm (t = 100 ms). Three thermal resistances: 10−5, 10−4 and10−3 m2W−1K are studied (Color figure online)

In Fig. 4 we show numerical simulations of the tempera-ture contrast as a function of time. Calculations are performedfor d = 1.0 mm and a = 0.75 mm. We have studied three dif-ferent values of Rth : 10−5, 10−4 and 10−3 m2W−1K. Theseresults indicate that as time goes by, the contrast increasesuntil it reaches a maximum value, which is higher for largerthermal resistances, in agreement with Fig. 2, and then startsdecreasing slowly for later times. Moreover, the maximumappears at longer times as the thermal resistance increases.

In Fig. 5 we show the dependence of �T on thelaser spot radius. Numerical calculations are performed ford = μ = 1.26 mm, which means t = 100 ms for AISI-304 samples. Three thermal resistances are studied: 10−5,10−4 and 10−3 m2W−1K. As can be observed, the maxi-mum temperature contrast corresponds to a ≈ 0.9 μ forthe thermal resistance of 10−3 m2W−1K, i.e., when the laser

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0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

T

d (

10-5

10-4

10-3

Fig. 6 Numerical simulation of �T as a function of the position thespot with respect to the crack. We have considered a cracked AISI-304sample with a = 0.9 μ. Three thermal resistances: 10−5, 10−4 and 10−3

m2W−1K are studied at t = 100 ms (Color figure online)

spot radius almost reaches the vertical crack. However, thisresult depends on the thermal resistance: the maximum of thecontrast is reached for lower radius spot sizes as the thermalresistance decreases. Nevertheless, there is a good contrastfor radii in the range 0.5–1.5μ.

Finally, in Fig. 6 we analyze the dependence of �T onthe distance of the laser spot to the crack. Calculations areperformed for fixed laser beam radius a = 0.9μ and measure-ment time t = 100 ms. We consider three values of thermalresistance Rth : 10−5, 10−4 and 10−3 m2W−1K. As can beobserved, the maximum temperature contrast is producedwhen the laser spot impinges the sample surface at d ≈0.75 μ and it quickly decreases as the laser spot moves awayfrom this position. Note that this result is valid for all Rth

values. On the other hand, the relative contrasts for d = μ

are equal to those ones obtained in Fig. 5 for each thermalresistance, as expected.

The results of this section can be summarized as follows:

(a) For a given thermal resistance, the largest temperaturecontrast at the crack position is obtained for d ≈ 0.75 μ

≈ 0.85a, i.e. when the laser spot slightly overlaps thecrack. However, in order to avoid light entering the crackand remembering that the maximum of �T as a func-tion of a is not very sharp, we propose the followingexperimental conditions:

d ≈ 0.75μ ≈ 1.1a. (12)

(b) In order to detect very narrow cracks, early times afterthe Dirac heating pulse should be used. This conditiontogether with Eq. (12) implies using a tightly focusedlaser spot very close to the crack. For instance, usingd = 0.1 mm on metallic samples together with t ≈ 1 msallows to detect thermal resistances as low as 10−6–10−7

m2W−1K. Note that these experimental conditions arealso valid to detect thicker cracks, but not size themaccurately, since the temperature contrast saturates.

(c) In order to retrieve Rth accurately, measurement timesshowing a contrast about half of the maximum one areproposed since they are the most sensitive to Rth varia-tions.

4 Experimental Results

Figure 7 shows the diagram of the experimental set-up. Wehave used a pulsed Nd-Glass laser at 1.053 μm with “flat-top” spatial profile and adjustable energy (up to 25 J/pulse)as the heating source. The time duration of the pulse is about0.4 ms (see Fig. 2 in [28]), so we can consider it as instanta-neous (Dirac-like pulse). The positive lens focuses the laserbeam and a Ge window directs it to the sample surface. ThisGe window also prevents the laser radiation from reachingthe camera lens. An infrared microscope lens collects the IRradiation emitted by the sample and we record it with an IRcamera (FLIR SC7500 working between 3 and 5 μm). Thespatial resolution of this IR system is such that every pixelaverages the infrared emission from a 31 μm side square.We record the images at 200 images/s frame rate. The noise

Fig. 7 Diagram of theexperimental set-up used for thecharacterization of verticalcracks

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J Nondestruct Eval (2016) 35:22 Page 7 of 10 22

equivalent temperature difference (NETD) [29] of our IRcamera is about 20 mK. It is worth mentioning that the tem-perature detected by our IR camera is not the real surfacetemperature rise of the sample, since neither its absorptivityto the laser wavelength nor its IR emissivity is known. How-ever, the “apparent” temperature rise �T measured by the IRdetectors is proportional to the real temperature rise of thesample surface.

According to Eq. (10), the spatial profile of the tempera-ture at each measurement time depends on five parameters(Qo/e,α, K Rth , a and d). However, in the experimental mea-surements we fix the distance between the laser spot and thecrack position (d ≈ 0.8–1 mm) in order to improve the sen-sitivity and the contrast as described in Sect. 3. On the otherhand, the radius of the laser spot (a) and the thermal dif-fusivity of the sample (α) are measured far away from thecrack position by making a curve fitting of the temperatureprofile along the y-axis to Eq. (4) at several times. In fact,Eq. (4) indicates that the temperature profile has a Gaussianprofile with radius b = a2 + 8αt. Accordingly, by measur-ing b at several times a straight line is obtained whose slopegives α and the intercept provides a [24]. Finally, the onlyparameters involved in the curve fittings of the temperaturey-profiles using Eq. (10) are K Rth and Qo/e.

We have prepared calibrated vertical cracks using twoblocks of AISI-304 stainless steel 2 cm thick. As this metallicalloy has a shiny surface, a thin graphite grey layer of about3 μm thick has been sprayed onto the surface to increaseboth the absorption to the heating laser and the emissivityto infrared wavelengths. In order to calibrate the air gapbetween the two AISI-304 blocks, we placed nickel tapesof 25, 10, 5, 2.5 and 1 μm thick between them, which pro-duces thermal resistances of 10−3, 4×10−4, 2×10−4, 10−4

and 4×10−5 m2W−1K, respectively (Rth = L/Kair , Kair =0.026 Wm−1K−1).

In Fig. 8a we show the thermogram corresponding to a 1μm thick crack, obtained 70 ms after the heating pulse. Ascan be seen, even such a small air gap is clearly visible in thethermal image. Following the method described above we

Table 1 Results of the crack thickness obtained for AISI-304 and PEEK

Calibratedthickness (μm)

AISI-304 crackthickness (μm)

PEEK crackthickness (μm)

1 0.74 ± 0.05 –

2.5 1.9 ± 0.1 –

5 5.4 ± 0.3 –

10 17 ± 3 10 ± 1

25 28 ± 4 24 ± 3

50 – 48 ± 4

100 – 90 ± 5

measured a = 0.78 ± 0.04 mm and α = 3.7 ± 0.08 mm2s−1,which were kept constant for all crack thicknesses. On theother hand, we chose d = 0.81 mm for all cracked samples.Figure 8b shows the temperature profiles along the y-axisfor the vertical cracks at t = 70 ms after the laser pulse (i.e.μ = 1 mm, satisfying Eq. 12). The temperature values areshifted in order to better show the jump at the crack position.Dots represent the experimental data and the continuous linethe fit to Eq. (10). A non-linear least square fit based on theLevenberg–Marquardt algorithm was implemented [30,31]with two free parameters (K Rth and Qo/e). Table 1 showsthe results obtained for the cracks thicknesses. Notice thegood agreement between the estimated values and the tapesthicknesses. We fitted data collected at five times, from 70to 110 ms, to obtain the standard deviation of the estimatedvalues.

Since narrow cracks are harder to characterize in ther-mal insulator materials, we performed measurements oncalibrated cracks using two plates of polyether-ether-ketone(PEEK) 1 cm thick. As this polymer is semitransparent, a thinmatt black synthetic enamel layer of about 5 μm thick hasbeen sprayed on the surface to make it opaque to the excit-ing wavelength. In this case, we placed nickel tapes of 10(4×10−4), 25 (10−3), 50 (2×10−3) and 100 (4×10−3) μm(m2W−1K) thick (thermal resistance). In Fig. 9a we show the

Fig. 8 a Thermogram for twoAISI-304 blocks put in contactto simulate an infinite verticalcrack 1 μm thick at t = 70 ms.We used the followingexperimental parameters:d = 0.81 mm, a = 0.78 mm andα = 3.7 mm2s−1. b Temperatureprofiles along the y-axis forseveral crack widths att = 70 ms. Dots correspond toexperimental data andcontinuous lines to the fit to Eq.(10) (Color figure online)

0

1

2

3

4

-2 -1 0 1 2 3

T (

K)

y (mm)

1 m2.5 m

5 m

10 m

25 m(b)

(a)

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Fig. 9 a Thermogram for twoPEEK plates put in contact tosimulate an infinite verticalcrack 10 μm thick at t = 500 ms.We used the followingexperimental parameters:d = 0.94 mm, a = 0.93 mm andα = 0.19 mm2s−1. bTemperature profiles along they-axis for several crack widths.Dots correspond to experimentaldata and continuous lines to thefit to Eq. (10) (Color figureonline)

0

1

2

3

4

5

6

-2 -1 0 1 2 3 4

T (

K)

y (mm)

10 m25 m

50 m

100 m

(b)(a)

thermogram corresponding to a 10 μm thick crack, obtained500 ms after the heating pulse. Note that this 10 μm thickcrack is more difficult to distinguish than the 1 μm thick crackin AISI-304, in agreement with the theoretical predictions.In fact, the product K Rth is higher for a 1 μm thick crack inAISI-304 (6×10−4 m) than for a 10 μm thick crack in PEEK(1×10−4 m). The measured values of the experimental para-meters were: α= 0.190 ± 0.004 mm2s−1,a = 0.77 ± 0.05 mmand d = 0.84 mm. Notice that for the 100 μm thick crack,we chose d ≈ 1 mm. This is because for smaller valuesof d, the laser light comes into the ‘air gap’ and the pro-posed model does not hold. Figure 9b shows the temperatureprofiles along the y-axis for all the vertical cracks in PEEK.As before, the values of the temperature are shifted in orderto show the jump at the crack position. We use dots for theexperimental results and continuous lines for the fittings toEq. (10). Table 1 shows the results obtained for the cracksthicknesses. Similarly to AISI-304 cracked samples, there isa good agreement between the estimated values and the tapesthicknesses in PEEK. In this case, the uncertainty is the stan-dard deviation corresponding to seven measurement times,from 500 to 800 ms.

Finally, we put the two AISI-304 blocks in direct contact,i.e. without nickel tape between them. As the surfaces in con-tact are polished, they simulate an extremely thin crack. Theresult depends slightly on the position in which the sample isexcited, probably due to the different edge surface conditions.This result allows us to conclude that the upper limit for thisthermal resistance is Rth ≤ 2.3×10−5 m2W−1K (Lcrack ≤600 nm). Similarly, for the PEEK plates, we found that theupper limit for this thermal resistance is Rth ≤ 3.1×10−4

m2W−1K (Lcrack ≤ 8 μm).Note that in the experiments we used a ‘flat-top’ instead

of the Gaussian laser assumed in the theory. Although atshort times after the heating pulse, both spatial temperaturedistributions are very different, we verified numerically, thatas times goes by, both temperature fields converge. In fact,when the thermal diffusion length verifies μ = √

4αt ≥1.6a, the difference between both temperatures is smaller

than 1 % at each point. This means that for times satisfyingt ≥ 2.56a2

4α, predictions from a Gaussian laser and a “flat-top”

one are indistinguishable. In order to fulfill this condition,we chose t ≥ 70 ms for AISI-304 and t ≥ 500 ms forPEEK in the measurements.

It is worth noting that the experimental results shown inFigs. 8b and 9b do not exhibit a sharp discontinuity at thecrack, but a smooth transition involving around 4 pixels. Thisresult is due to the imperfect imaging system of the IR cam-era (diffraction, multiple reflections, flare…). The so-calledPoint Spread Function (PSF) of the optical system quantifiesits effect, which depends on the lens quality [32]. As in oursystem the effect is quite small, we have not taken it intoaccount in the fittings.

Equation (10), which has been used for fittings, is onlyvalid for infinite vertical cracks. All simulations and experi-ments shown in this work were performed on infinite cracks.However, real cracks have a limited area. For finite cracksthere are no semi-analytical solutions, and sophisticatednumerical methods must be used [20] that are outside thescope of this work. Nevertheless a practical question arises:which is the minimum size of a crack whose width could beobtained accurately using Eq. (10)? A rough answer to thisquestion is that both length and depth of the crack shouldbe higher than (at least twice) the thermal diffusion length.This means that tightly focusing the laser beam very closeto the crack and using short times after the laser pulse, sub-milimetric cracks could be characterized. However, a tightlyfocused laser pulse could easily damage the sample. Keepingthe experimental data used in this work, a ≈ 0.8 mm andd ≈ 0.9 mm, together with the optimal contrast conditionproposed at the end of Sect. 3, d ≈ 0.75μ ≈ 0.85a, a morerealistic conclusion is that the model developed in this workcan be used to size the width of open cracks larger than 4 mmin length and 2 mm in depth.

Before concluding this section, let us make some com-ments on the time consumption of the method. Once the laserradius is measured and the laser spot is placed close to thecrack, the acquisition time of the temperature evolution of

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the sample surface after the laser pulse is about 2 s. Then, thedata processing to obtain the temperature profile across decrack at a given time after the laser pulse takes about 3 minsince it is done manually (15 min for five profiles). Finally,the fitting procedure to retrieve the crack width takes only1 min. This means that the total duration could be drasticallyreduced by automating the data selection and processing pro-cedure. Thus, a fast characterization of infinite vertical crackswith pulsed laser spot thermography could be implementedusing the method presented in this work.

5 Conclusions

In this work, we have dealt with the thickness characteriza-tion of infinite vertical cracks using pulsed laser spot infraredthermography. First, we have found a semi-analytical expres-sion for the surface temperature of a material containing sucha crack when a Dirac-like pulse laser beam of Gaussian shapeimpinges close to the crack. The presence of the crack pro-duces an abrupt jump in the surface temperature at the crackposition. Numerical simulations indicate that the highest tem-perature contrast is produced when a, d and μ satisfy d ≈0.75μ ≈ 0.85a. However, in order to avoid the laser pulsefrom entering the crack we have used the following exper-imental rule: d ≈ 0.75μ ≈ 1.1a. The validity of the modelhas been tested by performing pulsed infrared thermographymeasurements on AISI-304 stainless steel and PEEK samplescontaining calibrated cracks. The thickness of the crack wasobtained by fitting the surface temperature along the profileperpendicular to the crack through the center of the Gaussianspot. The agreement between the optically calibrated widthand the retrieved one is very good even for widths as narrowas 1 μm in stainless steel and 10 μm in PEEK. More efforthas to be done in order to characterize semi-infinite and finitevertical cracks, which allows real application for industry.

Acknowledgments This work has been supported by the Ministe-rio de Ciencia e Innovación (MAT2011-23811), by Gobierno Vasco(IT619-13), by UPV/EHU (UFI 11/55), by CINVESTAV UnidadMérida and by CONACYT.

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