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Applied Thermal Engineering 61 (2013) 596e605

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Application of 3D heat diffusion to detect embedded 3D empty cracks

C. Serra a, *, A. Tadeu b, J. Prata b, N. Simões b

a ITeCons e Instituto de Investigação e Desenvolvimento Tecnológico em Ciências da Construção, Rua Pedro Hispano s/n, 3030-289 Coimbra, Portugalb Department of Civil Engineering, University of Coimbra, Pólo II, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal

h i g h l i g h t s

� 3D heat diffusion by conduction around a 3D thin crack is modeled.� A boundary element method in the frequency domain is used to simulate heat diffusion.� Boundary element is formulated in terms of normal-derivative integral equations (TBEM).� Phase-contrast images for defect detection using infrared thermography are computed.� Crack size, shape, position (depth, inclination) and distance to source is studied.

a r t i c l e i n f o

Article history:Received 8 October 2012Accepted 27 August 2013Available online 5 September 2013

Keywords:3D heat sourcesInfrared thermographyNormal-derivative integral equations(TBEM)Transient heat diffusionPhase-contrast

* Corresponding author.E-mail addresses: cserra@itecons.uc.pt, inaserra@

dec.uc.pt (A. Tadeu), jprata@itecons.uc.pt (J. Prata), nas

1359-4311/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.applthermaleng.2013.08.03

a b s t r a c t

This paper presents a 3D boundary element model (BEM), formulated in the frequency domain, tosimulate heat diffusion by conduction in the vicinity of 3D cracks. The model intends to contribute to theinterpretation of infrared thermography (IRT) data results and to explore the features of this non-destructive testing technique (NDT) when it is used to detect and characterize defects. The defect isassumed to be a null thickness crack embedded in an unbounded medium. The crack does not allowdiffusion of energy, therefore null heat fluxes are prescribed along its boundary. The BEM is written interms of normal-derivative integral equations (TBEM) in order to handle null thickness defects. Theresulting hypersingular integrals are solved analytically.

The applicability of the proposed methodology to defect detection tests is studied once the TBEMresults have been verified by means of known analytical solutions. Heat diffusion generated by a 3D pointheat source placed in the vicinity of a crack is modeled. The resulting thermal waves phase is comparedwith that obtained when the defect is absent, so as to understand the influence of crack characteristics onthe IRT data results analysis, especially on the phase-contrast images. Parameters such as the size of thecrack, its shape, its position (buried depth and inclination) and its distance from the heat source areanalyzed. Some conclusions are drawn on the effects of varying those parameters.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Heat and moisture diffusion is sensitive to the presence of de-fects. The analysis of thermal pattern images collected via infraredthermography (IRT) has therefore proven to be a useful non-destructive testing(NDT) technique for defect detection in a widerange of applications and sectors. NDT involving IRT can be per-formed by taking a passive approach if the materials and structuresare tested in their natural state. If relevant thermographic data arenot obtained an active approach can be taken, in which an

gmail.com (C. Serra), tadeu@imoes@dec.uc.pt (N. Simões).

All rights reserved.7

additional heat source is used to generate the temperature differ-ence which is not produced otherwise [1]. Thermal imaging isgenerally used in building diagnostics for detecting heat loss,missing or damaged insulation, thermal bridges, air leakages andexcess moisture [2]. However, since the properties of subsurfacedefects can be detected and their quantitative characterizationachieved by solving heat transfer problems using active IRT data,active IRT has been extensively used in NDT tests in a wide range ofapplications, including civil engineering [3,4].

Different types of active IRT can be used, depending on how thesurfaces are stimulated to produce a temperature gradient in thetest specimen [5]. One type is lock-in thermography, where boththermal and phase images’ results are recorded. Phase images areless influenced by surface characteristics and less sensitive to non-uniform heating and do not require the previous knowledge of the

List of acronyms

3D Three-dimensionalBEM Boundary element methodIRT Infrared thermographyNDT Non-destructive testingREC Receiver placed in a specific location on the grid of

receivers G1REF Case of an infinite sized crack with null heat fluxesTBEM Normal-derivative integral equations formulation

of the boundary element method

Fig. 1. 3D geometry of the problem.

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605 597

position of non-defect areas that is required in simple thermalcontrast processing.

According to Maldague [5] comparisons between numericalmodels and experimental results are not straightforward, but suchmodeling gives useful information about the general thermalbehavior of the sample, and it is useful to establish the limits of theeffectiveness of IRT and set inspection parameters. Furthermore, itallows the consideration of different defect geometries, to deter-mine their detectability and the expected optimum observationtime window for the best subsurface defect visibility without theexpense of making and testing specimens. This attests to the in-terest in solving heat transfer problems for the detection andquantitative characterization of the properties of subsurface de-fects. However, the 3D nature of subsurface defects combined withthe need to simulate heat transfer and diffusion phenomena in atransient regime presents a challenge for researchers [6,7].

Of the available numerical methods for homogeneous un-bounded or semi-infinite systems modeling, the BEM is the onethat automatically satisfies far field conditions and therefore onlyrequires the discretization of the inclusion’s boundaries [8]. Unlikedomain-discretization based techniques which give sparse systemsof equations, the BEM allows a compact description of the regions,resulting in fully populated systems of equations. The majordrawbacks of the BEM are that it requires the prior knowledge ofthe fundamental solution, and that it may lead to singular or nearlysingular integrals. However, it is still considered to be one of thebest tools for this kind of problem.

When the thickness of the heterogeneity being modeled tendstowards zero, as in the case of cracks or thin defects, the conven-tional direct BEM degenerates and becomes inaccurate. Varioustechniques have been proposed to overcome this, such as thenormal-derivative integral equations formulation (TBEM), whichleads to hypersingular integrals. Various numerical methods havebeen proposed to overcome difficulties posed by such singularities[9e13].

This paper describes a model motivated by an interest inassessing the potential of using IRT in the presence of defects withspecific geometries and depths. In the sections that follow, theproblem is first defined and the TBEM formulated in the frequencydomain is presented. Analytical solutions are used to solve thehypersingular integrals that appear in the 3D TBEM formulation,when the element being integrated is the loaded one. Finally, wedescribe the methodology used to obtain time-domain responsesfrom frequency-domain calculations, followed by a number of nu-merical applications to illustrate the applicability and usefulness ofthe proposed approach in the analysis of several test cases, with thesimulation of heat diffusion generated by a point heat source placedin the vicinity of a crack. These applications focus mainly on theeffects on the resulting diffused heat field and on phase contrastresults of changing the size of the crack, its shape, its position(buried depth and inclination), and the position of the 3D source.

2. Methods

2.1. Problem definition

This work simulates the three-dimensional heat diffusion in thevicinity of a 3D crack, generated by a point heat source, as illus-trated in Fig. 1. The medium is an unbounded spatially uniformsolid medium of density r, thermal conductivity l and specific heatc. The crack, with surface S, is assumed to have null thickness. Nullheat fluxes are prescribed along the surface.

This system is subjected to a point heat source placed at (xs, ys zs),

bf ðx; y; z; tÞ ¼ Pdðx� xsÞdðy� ysÞdðz� zsÞeiut (1)

where d (x � xs), d (y � ys) and d (z � zs) are Dirac-delta functions,and u is the frequency of the source. The response of this heatsource can be given by the expression:

Tincðx; y; z; xs; ys; zs;uÞ ¼ Pe�iffiffiffiffiffiffi�iu

K

pr0

2lr0(2)

in which K is the thermal diffusivity defined by l/r c, u is theoscillating frequency, i ¼

ffiffiffiffiffiffiffi�1

p,

r0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xsÞ2 þ ðy� ysÞ2 þ ðz� zsÞ2

qand P is the amplitude of

the heat source.

2.2. The normal-derivative integral equation (TBEM)

The transient heat transfer by conduction is governed by thediffusion equation:

v2

vx2þ v2

vy2þ v2

vz2

!Tðx;uÞ þ ðkcÞ2Tðx;uÞ ¼ 0 (3)

in which x ¼ (x, y, z) and kc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�iu=K

p.

The classical boundary integral equation, formulated in thefrequency domain, is:

bTðx0;uÞ ¼ �ZS

Hðx;nn1; x0;uÞTðx;uÞ dsþ Tincðx0; xs;uÞ

(4)

where H are the fundamental solutions (Green’s functions) for theheat fluxes, at a point x ¼ (x, y, z) on the boundary of the surface S,due to a virtual point heat source at x0 ¼ (x0, y0, z0); nn1 representsthe unit outward normal along the boundary S, at x ¼ (x, y, z); b is aconstant defined by the shape of the boundary, taking the value 1/2if x0 ¼ (x0, y0, z0)˛S, and 1 otherwise.

Fig. 2. 3D view of a 3D null thickness crack hosted in an unbounded solid medium.

Fig. 3. Geometry of the systems modeled: a) vertical cross section; b) front view withplacement of receivers REC A to REC F.

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605598

The required Green’s functions for heat flux in an unboundedmedium, in Cartesian coordinates, are given by:

Hðx;nn1; x0;uÞ ¼ e�ikcrð�ikcr � 1Þ4lpr2

vrvnn1

(5)

with r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x0Þ2 þ ðy� y0Þ2 þ ðz� z0Þ2

q.

The normal-derivative integral equation can be derived byapplying the gradient operator to the boundary integral equation(4), which can be seen as assuming the existence of dipole heatsources. When the boundary of the inclusion is loaded with dipoles(dynamic doublets) the required integral equations can beexpressed as:

a Tðx0;uÞ ¼ �ZS

Hðx;nn1;nn2; x0;uÞTðx;uÞ ds

þ T incðx0;nn2; xs;uÞ (6)

The Green’s functions H are defined by applying the gradientoperator to H, which can be seen as the derivatives of these formerGreen’s functions, to obtain heat fluxes. In these equations, nn2 isthe unit outward normal to the boundary S at the collocation pointsx0¼ (x0, y0, z0), defined by the vector nn2. In this equation, the factora is null for piecewise planar boundary elements.

The required three-dimensional Green’s functions for an un-bounded space are now defined as:

Hðx;nn1;nn2; x0;uÞ ¼ vHvx

vxvnn2

þ vHvy

vyvnn2

þ vHvz

vzvnn2

(7)

with

vHvx

ðx;nn1;nn2; x0;uÞ ¼ 14p

�A��

vrvx

�2 vxvnn1

þ vrvx

vrvy

vyvnn1

þ vrvx

vrvz

vzvnn1

�þ B�

vxvnn1

��;

vHvy

ðx;nn1;nn2; x0;uÞ ¼ 14p

�A�vrvx

vrvy

vxvn

þ�vrvy

�2 vyvn

n1 n1

þ vrvy

vrvz

vzvnn1

�þ B�

vyvnn1

��;

vHvz

ðx;nn1;nn2; x0;uÞ ¼ 14p

�A�vrvx

vrvz

vxvnn1

þ vrvy

vrvz

vyvnn1

þ�vrvz

�2 vzvnn1

�þ B�

vzvnn1

��;

A ¼ �k2c e�ikcr

rþ 3ikce�ikcr

r2þ 3e�ikcr

r3and B

¼ �ikce�ikcr

r2� e�ikcr

r3:

Fig. 4. Snapshots of temperature distribution, in �C: a) cracked medium; b) medium empty of cracks; c) temperature differences between results in a) and b).

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605 599

In equation (6) the incident heat field is computed as

Fig. 5. Phase-contrast curves obtained in REC A for a 0.2 � 0.2 m2 vertical crack placedat x ¼ 0.6 m, and for an infinitely sized crack (curve REF), subjected to a 3D pointsource at the origin.

T incðx;nn2; xs;uÞ ¼ Pe�ikcr0ð � ikcr0 � 1Þ2r20

�vr0vx

vxvnn2

þ vr0vy

vyvnn2

þ vr0vz

vzvnn2

�(8)

The global solution is found by solving equation (6), which re-quires the discretization of the interface S into N planar boundaryelements, with one nodal point in the middle of each element. Theintegrations in equation (6) are evaluated using a Gaussian quad-rature scheme when the element to be integrated is not the loadedelement. For the loaded element (the hypersingular element),however, the integrands exhibit a singularity and the integrationcan be carried out in closed form, as defined by Tadeu et al. [14].

The proposed algorithmwas verified using a circular cylindricalcavity for which analytical solutions are known. The results showedgood approximation between the responses calculated analyticallyand those given by the BEM formulation, and the solution improvedfor higher numbers of boundary elements. This verification is notgiven here for reasons of space.

3. Results and discussion

3.1. Case studies

The applicability and usefulness of the proposed approach areillustrated by simulating the heat propagation around a 3D nullthickness crack hosted in an unbounded solid medium andcomputing the phase-contrast images of the resulting heat wave.Various parameters are changed for the study of the influence ofthe properties of the plane crack S (see Fig. 2) on heat diffusion andphase image computations. The parameters changed are the shapeof the crack, its size, position and distance from the heat source andto the surface (grid of receivers).

Initially, the crack is parallel to the yz plane and the 3D pointsource is located at its origin (xs, ys, zs) ¼ (0.0 m, 0.0 m, 0.0 m).Computations are performed over three fine grids of receivers. Twogrids are placed at a distance from the source, along the x axis: gridG1 is 0.0675 m from the crack and is placed over a plane atx¼ 0.5325 m, and grid G2 is placed over the plane at x¼ 0.6725 m.The third grid of receivers, G3, is parallel to the z axis over the planelocated at z¼ 0.0 m. The receivers were spaced at equal intervals of0.005 m, 0.00625 m and 0.006 m in the x, y and z axis direction,

respectively. The grid of receivers G1 simulates an object’s surfaceon which thermographic data given by IRT is recorded.

Three different geometries are simulated in order to study theinfluence of the size of the crack (see Fig. 3b). Additionally, a0.05 � 0.2 m2 crack is modeled to study the influence of changingthe shape of the crack. The crack is initially placed at x ¼ 0.6 m.Changes produced when the crack is closer to the simulated ob-ject’s surface were studied by moving the grid of receivers 0.04 mcloser to the crack. The influence of changing the distance fromsource to defect was studied bymoving the source to x¼ 0.4525 m.The influence of the crack inclination was studied by tilting thevertical crack around an axis parallel to the z axis, by a ¼ 15� anda ¼ 40� (see Fig. 3a). Results are analyzed for specific receiversplaced at the center and the extremities of the cracks, as shown inFig. 3b. Responses are also computed for the example of an infinitesized crack with null heat fluxes (labeled REF) by applying an imagesource technique.

3.2. Simulation results of heat field in the time domain and thermalwave phase

The heat field in the time domainwas determined by applying anumerical inverse fast Fourier transform in the frequency domain.Aliasing phenomena are dealt by introducing complex frequencieswith a small imaginary part, taking the form uc ¼ u � ih (where

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605600

h ¼ 0.7Du, and Du is the frequency step), following the techniquedescribed in Ref. [15].

The thermal properties of the host medium are those of concrete:c¼ 880 J kg�1 �C�1, r¼ 2300 kgm�3 and l¼ 1.40 Wm�1 �C�1. Allthe calculations are performed in the [0.0,1.024�10�3] Hz frequencyrange, with an increment of Df ¼ 0.5 � 10�6 Hz, resulting in a 1/(0.5�10�6) s total timewindowfor theanalysis. Theupper limitof thefrequency range was defined so as to ensure a negligible contributionto the responses. Responses for higher frequencies would have a nullcontribution. The imaginary part of the frequency is given byh¼ 0.7� 2pDf. The hostmedium is assumed to be initially at 20.0 �C.The source starts emitting energy at instant t¼ 0.5 h and it continuesfor 1.5 h. The heat source time dependence is assumed to be rectan-gular with an amplitude of P ¼ 154.5. This amplitude was defined sothatamaximumtemperature increaseof15.0 �C is registeredatRECA.

The effects of the presence of a defect on the temperaturediffusion pattern, computations were studied for a mediumwithout any cracks and a medium with an embedded crack. Thedifference between the two results expresses the differences intemperature that can be used by IRT in defect detection. Fig. 4 il-lustrates the time domain snapshots of the temperature distribu-tion at t ¼ 20 h for a 0.2 � 0.2 m2 vertical crack placed atx ¼ 0.6 m, subjected to a 3D point source placed at the origin, onthe three grids of receivers placed at xG1 ¼ 0.5325 m,xG2 ¼ 0.6725 m and zG3 ¼ 0.0 m.

Thermal wave phase results are computed directly in the fre-quency domain, assuming the existence of harmonic sources with

Fig. 6. Phase-contrast curves obtained for varying crack sizes: a) in REC A (x ¼ 0.5325 m, y(x ¼ 0.5325 m, y ¼ 0.0 m, z ¼ 0.1 m); d) in REC D (x ¼ 0.5325 m, y ¼ 0.0 m, z ¼ 0.15 m).

unitary amplitude. This corresponds to an ideal Dirac pulse in thetime domain, which has an infinite flat spectrum in the frequencydomain. Phase-contrast images are obtained by computing thedifference between the phase of the heat waves within themediumwhen the crack is embedded in the medium and for the mediumwithout any defects.

Fig. 5 illustrates the phase-contrast curve obtained at REC A. Theresulting curve shows a clearly defined peak in the frequencyspectra at maximum phase-contrast jDfmaxj, which corresponds tothe characteristic frequency fch used in defect detection. Thedetection threshold is given by the blind frequency fb at zero phase-contrast. Fig. 5 also makes it clear that an infinitely sized crack leadsto lower characteristic frequency fch.

3.2.1. Influence of crack sizeThe following graphs show the phase contrast curves obtained for

four different crack sizes: 0.3 � 0.3 m2; 0.2 � 0.2 m2; 0.1 � 0.1 m2

and 0.05 � 0.2 m2 at receivers REC A, REC B, REC C and REC D.The results show that, at all receivers, the characteristic fre-

quency fch decreases and the corresponding maximum phase-contrast absolute valuejDfmaxj increases, for larger crack sizes. Itcan be seen that increasing the size of the crack leads to resultscloser to the reference curve, which was expected since the largerthe crack the closer it gets to the behavior of an infinite size crack. Inno case does the blind frequency fb change significantly and thedetection threshold does not change much with crack size varia-tion. The phase-contrast response is significantly reduced when the

¼ 0.0 m, z ¼ 0.0 m); b) in REC B (x ¼ 0.5325 m, y ¼ 0.0 m, z ¼ 0.05 m); c) in REC C

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605 601

receivers are placed beyond the crack limits, as shown in Fig. 6c forREC C (placed outside the limits of the0.1 � 0.1 m2 crack) and inFig. 6d for REC D (placed outside the limits of the 0.1 � 0.1 m2,0.05 � 0.2 m2 and 0.2 � 0.2 m2 cracks). For any crack size, whenreceivers are located further from the center of the crack there is adecrease in the fch values. It is also seen that the amplitude of thephase-contrast decreases when the geometry of the crack changesfrom 0.1 � 0.1 m2 to 0.05 � 0.2 m2 (see Fig. 6a and b).

Fig. 7 contains snapshots of the phase-contrast images computedalong the grid of receivers G1, for three different crack sizes. Thesnapshots are taken at the frequency fch which is the maximumphase-contrast jDfmaxj observed in receiver REC A, located at the

Fig. 7. Snapshots of the phase-contrast images D4 (f) obtain

center of each crack (indicated in these plots by a round red mark).The phase-contrast snapshots clearly demonstrate the ability todetect the presence of the crack and to assess its size. It is clearly seenthat outside the crack limits the phase-contrast tends to zero becausethe effect of the reflected heat field is diminished.

3.2.2. Influence of source positionThe graphs in Fig. 8 illustrate the influence of changing the

source position and show the phase-contrast results obtainedwhen the source is moved from the origin to a position closer to thedefect, to x ¼ 0.4525 m, so that the distance from the source to thedefect is reduced from 0.6 m to 0.1475 m.

ed at the grid of receivers G1 for different crack sizes.

Fig. 8. Phase-contrast curves for results in REC A (0.5325 m, 0.0 m, 0.0 m) and REC C (0.5325 m, 0.0 m, 0.1 m) when the heat-source distance is reduced: a) 0.2 � 0.2 m2 size crack;b) infinite crack (REF).

Fig. 9. Phase-contrast modeling results in REC A, REC C, REC A0 , REC C0 for: a) 0.2 � 0.2 m2 crack; b) infinite crack (REF).

Table 1Product between the squared defect depth and the characteristic frequenciesrecorded at specific receivers.

Crack size

d2 � fch 0.1 � 0.1 m2 0.2 � 0.2 m2 0.3 � 0.3 m2 REF

REC A 2.51 � 10�7 1.69 � 10�7 1.23 � 10�7 0.57 � 10�7

REC A0 1.54 � 10�7 0.87 � 10�7 0.60 � 10�7 0.58 � 10�7

REC E 2.19 � 10�7 1.69 � 10�7 1.28 � 10�7 0.57 � 10�7

REC E0 1.23 � 10�7 1.05 � 10�7 0.71 � 10�7 0.59 � 10�7

REC B 2.30 � 10�7 1.66 � 10�7 1.25 � 10�7 0.57 � 10�7

REC B0 0.06 � 10�7 0.96 � 10�7 0.67 � 10�7 0.58 � 10�7

REC C 1.85 � 10�7 1.57 � 10�7 1.28 � 10�7 0.59 � 10�7

REC C0 0.04 � 10�7 0.89 � 10�7 0.86 � 10�7 0.59 � 10�7

REC D 1.41 � 10�7 1.30 � 10�7 1.23 � 10�7 0.62 � 10�7

REC D0 0.42 � 10�7 0.46 � 10�7 0.86 � 10�7 0.61 � 10�7

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605602

The blind frequency fb does not suffer significant alteration inany of the cases. When the distance is shorter there is a decrease inthe characteristic frequency fch value in REC A, and a decrease in thecorresponding maximum phase-contrast absolute value jDfmaxj.However, in REC C there is an increase in both the fch and jDfmaxj.The reference curves in Fig. 8b show that changing the source po-sition gives different results depending on the position of thereceiver, and that it has significant importance on the peak phase-contrast value, especially for receivers closer to the center of thecrack. For both cracks (0.2 � 0.2 m2 and infinite) the characteristicfrequency increases the further the receiver is from the center.

3.2.3. Influence of crack depthThe grid of receivers G1 is shifted along the x axis to study the

influence of the depth of the defect. The graphs in Fig. 9a show thephase-contrast results obtained from modeling a 0.2 � 0.2 m2

crack for receivers placed atxG1 ¼ 0.5325 m (REC A and REC C) andat xG1 ¼ 0.5725 m (REC A0 and REC C0), simulating a defect depthreduction. Fig. 9b shows the reference curves for the samepositions.

For both sizes results show that reducing the depth of the defectincreases the fch and the corresponding jDfmaxj. Furthermore, asthe buried depth is lessened the detection threshold given by theblind fb frequency also increases. The reference curves in Fig. 9b

show that for an infinite size crack the position of the receivers onthe grid does not influence results.

The resulting curves are analyzed further, aiming to inve-stigate the relationship between the depth of the defect and thecharacteristic frequency. The results shown in Table 1 relate to theproduct of the square of the defect depth d2 and the characteristicfrequency fch recorded at the receivers placed at xG1¼0.5725m (RECA0 to E0), for a defect depth of d ¼ 0.0275 m, and those originally

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605 603

placed at xG1 ¼ 0.5325 m (REC A to E), corresponding to an initialdefect depth of d ¼ 0.0675 m.

It can be seen that fairly similar results are obtained for aninfinite sized crack. In this case, the relationship between the depthof the defect d and the characteristic frequency fch can be approx-imately expressed by the equation given by Arndt for Civil Engi-neering applications of square-pulse thermography [16]:

d ¼ kc �ffiffiffiffiffiffiffiffiffiffiffiK=fch

q(9)

where kc is a correction factor dependant on the thermal propertiesof the medium, K is the thermal diffusivity as defined previouslyand fch is the characteristic frequency.

Fig. 10. Snapshots of phase-contrast images D4

Table 1 shows that results for finite size cracks get closer tothose for the infinite size crack the larger the size of the crack. Thecorrelation between d and fch is also closer to the one described inequation (9) for shorter defect depths and for receiver readingsclosest to the center of the defect.

3.2.4. Influence of crack inclinationThe snapshots in Fig. 10 show the phase-contrast image results

along G1 for three different crack positions: a ¼ 0�, a ¼ 15� anda ¼ 40�. The snapshots are taken at the frequencies correspondingto maximum phase-contrast in receivers REC E and REC F.

The images show that as the crack is tilted the peak phase-contrast occurs at higher frequencies and phase-contrast

(f) obtained for different crack positions.

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605604

amplitude is reduced at lower frequencies. Furthermore, as seenfor a ¼ 40�, in the low frequency spectrum the response isinverted and a positive difference is recorded in receivers locatedin front of the lower part of the crack. In these receivers the peakphase-contrast occurs for high fch values, which is agrees with theresults described in the previous section (reducing the depth ofthe defect increases fch and jDfmaxj values). However, as the crackis tilted further, the lower part of the defect gets closer to the

Fig. 11. Snapshots of temperature differences for a

surface and the upper part gets further away, which leads to amixed response. As shown by the images for a ¼ 40�, conclusionsregarding the size and shape of the crack can no longer bededuced by the phase-contrast image given at a specific fre-quency. We have to sweep a wide range of frequencies to fullyassess the size and shape of the defect.

To better understand these results, the heat field in the timedomain is computed via an inverse fast Fourier transform applied in

0.2 � 0.2 m2 crack tilted at different angles.

C. Serra et al. / Applied Thermal Engineering 61 (2013) 596e605 605

the frequency domain and by assuming the source time evolutiondefined previously. Computations are performed for a mediumwithout any cracks and a mediumwith the embedded 0.2� 0.2 m2

cracks. Fig. 11 shows snapshots of the temperature differences be-tween both results at different time instants (t¼ 10 h, t¼ 20 h andt ¼ 30 h) for different anglesa ¼ 0�, a ¼ 15� and a ¼ 40�.

Given the distance from the crack to the grid of receivers onx ¼ 0.5325 m, the temperature field is very mild for all cases. Re-sults show that at t ¼ 10 h the temperature is still increasing, whileat t ¼ 30 h the source has already been turned off for a long timeand the temperature is falling. The temperature field recorded onthe perpendicular grid G3 bears out this behavior (see Fig. 2).

The temperature field is progressively weaker as the crack istilted for the receivers in front of the upper part of the defect whileit is stronger for the lower receivers. This is particularly noticeableat t ¼ 10 h. When the crack is tilted horizontally the temperaturedifference is null (not shown), because the crack is aligned with thepoint heat source at z ¼ 0. In this specific case IRT analysis will notbe able to detect the defect.

4. Conclusions

In this paper the authors have proposed a three-dimensionalnormal-derivative integral equation (TBEM) formulation to allowheat conduction modeling in the vicinity of cracks embedded in anunbounded medium for applications using active IRT for defectdetection. The hypersingular integrals that appear when theelement to be integrated is the loaded one (singular element) areintegrated analytically. Our model is applicable to any medium thatcan be deemed homogeneous and where the heat conductionphenomena can be governed by the Helmholtz equation. Suchmedia may include steel, concrete, mortar, rock, polymers, etc.

The proposed formulation was employed to simulate the activeIRT procedure in a set of numerical examples where a crack isplaced in an unbounded space. Parameters which are relevantfeatures in non-destructive testing by active IRT, such as the depthat which the crack is embedded, its size, shape, inclination anddistance from the heat source, were analyzed.

Both the temperature differences and the phase-contrast imagesof the resulting thermal waves were computed by modeling theheat waves within the medium where there is an embedded crackand for the mediumwithout defects. Numerical results obtained inthis study demonstrate that the mentioned crack parameters mayhave considerable influence on active IRT data results. The nu-merical simulation results have proven the applicability of theproposed formulation to understanding heat diffusion, in both thetime and frequency domains, in the vicinity of 3D defects whensubjected to 3D sources. The results were found to be highlyinfluenced by the 3D nature of the crack and heat source. The heatdiffusion patterns generated by the proposed models may be founduseful for correctly interpreting experimental IRT results as well as

for successfully establishing experimental parameters for active IRTtesting.

Acknowledgements

The research work presented herein was supported by FEDERfunds through the Operational Programme for CompetitivenessFactors e COMPETE and by national funds through the FCT e Por-tuguese Foundation for Science and Technology, under researchproject PTDC/ECM/114189/2009 and was supported in part by theFCT doctoral grant SFRH/BD/91686/2012. This work has also beensupported by the Energy and Mobility for Sustainable Regions e

EMSURE e Project (CENTRO-07-0224-FEDER-002004).

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