PART 4. Thermal Tomography1
General Considerations brief initial thermal perturbation. It is
noticed that the occurrence of the time of
X-ray computed tomography (CT) is an maximum thermal contrast tcmax is
established technique used to reconstruct proportional to the square of the depth
the inner structures of components. Over (at least in homogeneous materials);
conventional X-radiographs1 it has the consequently deeper discontinuities will
experience longer fcmax· Thermal contrast
advantage of simplifying the C(i,j,t) is computed at time t for a given
interpretation of images because slices of pixel (i,j) from temperature image T:
the tested component can be sliced and
observed to any depth. This brings useful fiGURE 28. Moment of maximum contrast lcmax as function
depth information with respect to simple of subsurface discontinuity depth: (a) discontinuity depth
X-ray images on which all the inner diagram; (b) contrast.
structure information is compressed. In (a)
X-ray computed tomographic imaging a
set of X-radiographs is recorded along D--z,~
different projections around the tested
component and a special algorithm is z, +--+[]
then used to reconstruct the unique
distribution of attenuation coefficients -
inside the component.96-97 This map of
attenuation coefficients corresponds to z, 1-oE--~--~
the inner structure of the part expressed
as the ability of the matter to attenuate -
the rays.
(b)
Tomography cannot be applied directly
to the heat transfer process, which occurs
not in a straight direction but according
to a diffusion propagation scheme.
Nevertheless, the idea to slice the tested
component into different layers
corresponding to the distribution of
thermal properties at specific depth layers
is interesting. This idea was proposed by
Vavilov in 1986 and since then has been
explored b6' him with several research
groups.98-I 3
Instead of being based on angular
projections as for Xwray computed
tomography, thermal tomography is based
on the surface temperature evolution of
the tested component following the initial
thermal perturbation. As analogy with
Xwray computed tomography, time
increments may be associated with
angular projections. Thermal tomography
is just a different technique of processing
the thermogram sequence and presenting
the data.
Thermal Tomographic Timet (s)
Principles
The principle of thermal tomography can
be understood by drawing thermal
contrasts under a sample surface (Fig. 28).
Observation is in reflection following a
386 Infrared and Thermal Testing
T(i,j,t)- T(i,j,t~O) (such as maximum thermal contrast)
T,0 ,(t)- Tw,(t~o)
(28) C(i,j,t) extracted over the infrared image
sequence recorded during the thermal
nondestructive testing of a component.
where 7~0a corresponds to the surface Assuming uniform heating of the tested
temperature over a sound area in the
~mage. The image before heating (at t = O) surface, areas of the specimen having
IS also subtracted to remove spurious
thermal reflections. uniform thermal properties (such as
Using the classical test scheme of thermal resistance) will have the
surface flash heating with observation in occurrence of the parameter of interest
reflection, it is possible to measure the
temperature decay over the component (for example, maximum thermal contrast)
surface. From such a sequence of infrared
images recorded during the testing of the showing up in the same time window in
component, it is possible to compute, for the timegram. On the other hand,
every pixel in the field of view, the time
subsurface discontinuities having different
when, for a given pixel, the thermal
contrast Eq. 28 is maximum. This time is thermal properties will experience
called lcmax and the distribution of all the different values of the parameter of
~Cmax values f?r all the pixels in the image
IS called the tmwgram (TGcmax). Because a interest and consequently wHJ exhibit
timegram has the same dimensions as an
different time values in tile timegram.
infrared image, it can be displayed as an
image. Figure 29 illustrates the technique This yields to possible discontinuity
to compute timegram image TGcma .: detection.
For 'r:/i, i = 0,1, ...,(Maxrow-1); tor"''r:!j,
1\•fore interestingly, because the
j ~ 0, I,(Maxcol-1 ):
timegram TGcmax is the time distribution
of the occurrence of the maximum
thermal contrast computed for every pixel
in the field of view and because time of
maximum contrast tcmax is proportional
t? th: square ~f tl~e discontinuity depth,
tune mforrnatwn m such a timegram is
also indicative of the depth of subsurface
artifacts present in the component, if any.
To recover the depths of these artifacts it
is thus necessary to slice the timegram,'
that is to extract from the timegram the
From these definitions, it is understood values of time of maximum thermal
that a timegram is in fact an image of the
time distribution of a given parameter c,ontrast in a given time window ft1,t2j.
1 he timegram Slice TGcmax n t2 is
obtained by thresholding tilnegram
TGcmax in fixed gate time ft1,t2J:
For 'f;, I ~ 0, I, ..., (Maxrow 1)· for 'f
1' ~ 0, I, (Maxcol 1): I /'
~IGURE 29. Thermal tomography principle: computation of 0
trmegram TGcmax·
where lcmax (i,j) < t1;
Evolution of
thermal contrast
for pixel (i,J)
(31) SiiceTGcmax 11 ' t 2 (i,j) lcmax (i,i)
.'
Maximum
Infrared image /contrast where lt < tcmax (i,j) < tz;
sequence
'••~-_j- Time of
Image 1 maximum
contrast
Image 2 .(i,j)
Image 3 ''''''
where lcm 3x (i,j) > t2•
Thus Slice__TGcmMu t1, t2 corresponds
to a particular layer under sample surface
displaying the distribution of thermal
properties Zazt·ttzh-e1.cSourcrehssploicnediisngcadlleepdt h
ZJ·t,--1 to
a
thermal tmnogratll by analogy with
X~radiographic computed tomograms and
also because, from the timegram image,
the tested component can be sliced into
multiple thermal tomograms
Timegram TGcma. corresponding to different depth layers
under the surface.
Data Processing and Modeling for Infrared and Thermal Testing 387
It is important to notice that thermal limitation affects depth resolution in two
tomography can not see a subsurface ways: detector noise and surface noise.
discontinuity behind another, for the first From these considerations, depth
discontinuity prevents the thermal front resolution t..z can be estimated: Inl
from reaching the second discontinuity.
However, because the first discontinuity (37) '-z iltc_<_ ac
provokes concerns about the integrity of
the tested component, this limitation may ilz ilt
not necessarily be dramatic. Moreover,
thermal tomography can only be where Cnolse is the thermal contrast noise;
deployed in reflection (testing from one iJtc k·(i.lz)-1 is the variation of the
side), because in transmission, the time pafameter of interest with depth (k stands
information about discontinuity depth is for the parameter of interest- either
lost (roughly speaking, the time of arrival time of maximum contrast/ time of half
of the thermal front on the back surface is rise contrast or some other parameter);
the same whatever is the discontinuity and iJC.(dt)-1 is the first derivative of the
depth). thermal contrast curve. From Eq. 37, it is
seen that recourse to parameter t,_o.smax
The ability to isolate a specific depth or t,_]JBmax for Which ()C.(iJt)-l is greater
layer under surface sample tested with than for tc:max is more advantageous
thermographic nondestructive testing is because more layers can be resolved
the main attraction of this technique, because of the smaller .dz. As an example,
which requires no special apparatus. In in carbon epoxy it can be demonstrated
fact, this is only a different way to process that parameter tcJdBmax makes it possible
infrared images. In fact, from the to resolve layers at a depth more than
thermogram sequence recorded over a twice as great as that of parameter lcmax·
given tested component/ it is possible to
compute different types of timegrams Finally, depth resolution of thermal
(TGs) depending on the parameter of tomography can also be enhanced by
interest - either the time of maximum increasing the input energy power needed
contrast lcmax1 the time of half rise for component thermal stimulation
contrast tcJ/2max' the time of half decay because in this case Cnolse will be reduced.
contrast lct/2max or the time at 3 dB of Cue should be taken to avoid damaging
maximum contrast tc3dllmax: the sample through surface overheating if
a nondestructive test is still desired.
For 'Vi, i = 0, 1, ..., (Maxrow-1); for 'Vj, j
~ 0, 1, (Maxcol-1):
(33) TGc_mox(i,i) Thermal Tomographic
Results
(34) TGct,uwx(i,j) tc.}jmax(i,i)
Some thermal tomographic results are
(35) TGc_max j'i(i,j) lc_maxj~(i,j) shown in the case of a fluorocarbon resin
insert in a carbon epoX}' specimen
tc_JdBmax(i,j) (Fig. 30). A raw image recorded at the
time the contrast is maximum over the
where tc_max(l,j)' tc_t/2max!i,;J, tc_maxl/2!i,iJ discontinuity (lcmax == 4.35 s) is shown on
are extracted from the thermal contrast Fig. 30a. The smoothed maximum
evolution curve for pixel (i,j). Evaluation contrast image reveals a temperature
of parameters tc maxl/2 and tc_J/2maxU,j) is contrast over the discontinuity of about
often more reliable than evaluation of 6 percent (Fig. 30b). Noise reduction
tc max because of the slow variation of techniques can be used with profit at thi's
thermal contrast near its maximum value. step. Timegram TGc max synthesized from
60 images taken during the specimen's
An important objective in thermal cooling is shuwn on Fig. 30c: on this
tomography is to determine the depth of image, it is clearly seen that edge
resolution that ccm be achieved, that is, irregularities produce higher values of
/low uwny layers a given sample can be sliced fc max than a discontinuity area does. This
into (or a given thermal nondestructive testing dfScrepancy is hardly noticed at this
experiment? VaviloviOZ indicated that particular scale. Tomograms are shown on
depth resolution is limited both by the Fig. 30d to 30f. Separation of the layer 0.8
rate at which infrared images are recorded to 1.5 mm (0.03 to 0.06 in.), which
and also by temperature resolution of the presumably contains the discontinuity,
imager (limited by noise). In turn, noise shows the fluorocarbon resin insert very
distinctly (Fig. 30d).
A tomogram of layer 1.4 to 1.8 mm
(0.055 to 0.07 in.) reveals some deeper
disturbances around the discontinuity
388 Infrared and Thermal Testing
(Fig. 30e), although strictly speaking this tomogram clearly show~ the lack of major
could not be deeper as stated previously. discontinuities inside thi.s layer. _
It is probably just weaker inner An interesting addition to thermal
irregularities in tc max termS.
tomography is first to include a
Finally the layer 1.8 to 2.0 mm (0.07 to
0.08 in.) (Fig. 30f) shows a single signal discontinuity detection step that mnkes it
that can not be identified but this possible to remove unwanted structures
from the tomograms.
FIGURE 30. Thermal tomography of fluorocarbon resin insert in carbon epoxy specimen using
Tcmax parameter: (a) raw image; (b) orientation of dimensions in Figs. 30c to 30g;
(c) smoothed image; (d) timegram; (e) tomogram of layer from 0.8 to 1.5 mm (0.03 to
0.06 in.); (f) tomogram of layer from 1.4 to 1.8 mm (0.055 to 0.07 in.); (g) tomogram of
layer from 1.8 to 2.0 mm (0.7 to 0.08 in.). Specimen measures 4.25 mm (0.17 in.) thick,
28 layers black painted carbon fiber reinforced plastic panel with 10 mm (0.40 in.) diameter
fluorocarbon resin Implant inserted at eighth layer, 1.2 mm (0.047 in.) beneath front surface.
(a) (e)
(b) (f)
(g)
Data Processing and Modeling for Infrared and Thermal Testing 389
fiGURE 31. Thermal tomography of acrylic plate with four holes drilled from back surface at different depths:
(a) orientation of dimensions in Figs. 31 b to 30g; (b) summiltion of all raw images without correction for
radiometric distortion; (c) binary image following discontinuity detection; (d) timegram TGcmaxi (e) product
of image in Fig. 31 b and 31 c; (f) tomogram for depths smaller than 2 mm (0.08 in.); (g) only deepest
2.5 mm (0.1 0 in.) discontinuity visible in tomogram for depths greater than 2 mm (0.08 in.). Arbitrary
amplitude units.
(a) (e)
(b)
(f)
390 Infrared and Thermal Testing
Figure 31 illustrates the technique in
the case of thermal nondestructive testing
of an acrylic plate in which subsurface flat
bottom holes have been drilled at
different depths from the back surface.
The evaluation proceeds in two steps:
discontinuity detection and tomogram
computation. For discontinuity detection,
an algorithm is used: Figure 31b is the
summation image of all raw infrared
images (Eq. 7- notice uncorrected
radiometric effects on image edges)
·whereas Fig. 31c is an image of Fig. 31b
after segmentation. Figure 31d is the
timegram TGc max (Eq. 29). Figure 31e is
the product oCFig. 31c and 31d, which
makes it possible to suppress unwanted
regions of the timegram. Hnally Fig. 31f is
the tomogram for depth layer 0 to 2 mm
(0 to 50s) and Figure 31g is the
tomogram for depth layer >2 _mm
(>0.08 in.). In Fig. 31g the deeper 2.5 mm
(0.1 in.) discontinuity is separated from
three other shallower subsurface holes
(z11l2'h3e<rm2aml mto)m. ography offers another
way to look at thermal information in
terms of time- rather than of amplitude
as in standard thermal nondestructive
testing image processing. It has some
advantages concerning interpretation of
results especially because detected
structures appear directly in terms of
depth (after conversion from the time
domain). One drawback of the technique
is the amount of computation needed
because all images must be processed.
Other limitations are as follows: strong
reduction of spatial resolution with depth 1
detection limited to subsurface artifacts
having different thermal properties with
respect to the bulk of material. These
limitations are shared by the standard
thermal nondestructive test procedure as
weB.
Data Processing and Modeling for Infrared and Thermal Testing 391
PART 5. Photothe:rmal Depth Profiling by
Thermal Wave Backscattering
Photothermal techniques have been Therefore the photothermal signal
successfully applied as quantitative tools contains information on thermal depth
for nondestructive characterization of profiles, which may be reconstructed by
heterogeneous materials. Such techniques simply comparing theory with experiment
are widely used to evaluate different types for the surface temperature T~urf or for any
of heterogeneity- either macroscopic related photothermal signal.
subsurface discontinuities in a
homogeneous materiaJI04 or microscopic The theoretical value of T~urf may be
structural modifications that produce provided by different models of heat
local thermal conductivity and diffusivity diffusion in heterogeneous materials,
changes.ws, 106 when k(z) and D(z) are known (direct
problem). Unfortunately many rigorous
The present discussion focuses on models allow calculating 1~urf by using
materials 1Nhose surfaces are subjected to recursive or numerical algorithms, in
thermal (such as hardening),I06,I07 which a clear relationship bet\veen Tsurf
mechanical (such as grinding)10S or and the thermal parameters is lost.
chemical treatments- to induce Consequently the depth profile
structural modifications in near surface reconstruction (inverse problem) consists of
layers that may be described by a depth a huge set of attempts for fitting Ts111f by
dependence of the thermal conductivity trying all the reasonable profiles. Several
k(z) and diffusivity D(z). Photothermal procedures have been introduced in the
depth profiling is usually applied to past to optimize such a heuristic fitting
heterogeneous materials and permits the procedure:
reconstruction of the thermal
conductivity or diffusivity depth profUes 1. In the frequency domain a stepwise
by monitoring the photothermal signal in least squares fit has been used to
the fol1owing situations. reconstruct a polygonal best
approximation to the conductivity
1. For depth profiling in frequency profile,I06,J07,11 3 a neural network
domain, the specimen is illuminated
by a wide pump laser beam modulated approach has been used to find the
best fit, 114 an inverse procedure has
at an adjustable frequency f. The been used to find the taylor expansion
parameters of the conductivity
photothermal signal is measured profiles,1 15 an inverse Green's function
versus the frequency. technique has been usect,ll6,117 a
2. For depth profiling using lateral scan,
the specimen is illuminated by a Hamilton/jacobi based model has been
focused pump laser beam modulated used for weak scattering118,l 19 and a
at some frequency f. The thermal wave impedance based model
has been used. 120
photothermal signal is measured as a 2. In the spatial domain the inverse
function of the distance from the scattering technique has been used to
heating point. reconstruct both thermal conductivity
3. For depth profiling in time domain, and heat capacity depth profiles121
the specimen is Hluminated by a wide and the conjugate gradient technique
pump pulsed laser beam. The has been used to optimize the ftt.l 22
photothermal signal is measured as a 3. In the time domain the effusivity
function of the time delay from the depth profile has been
pulse. reconstructed123•124 and the neural
network approach has been used to
The general idea is to generate at the find the best fit. 125
surface thermal waves (cases 1 and 2
above) or a thermal pulse (case 3) by In the following discussion an
periodical or pulsed laser heating. The inversion procedure is based on the
thermal waves, or the pulse, penetrate thermal backscatterins model of heat
inside the sample, are subjected to conduction in a heterogeneous
backscattering due to thermal effusivity material. 126-130 In this approximate
changes and come back to the surface.
The surface temperature resulting from theoretical model the natural relationship
the superposition between the main field between T,11rf and the depth profiles is put
and the backscattered field is eventual~' into evidence, leading to a fundamental
detected by photothermal radiometry1 9 simplification of the inwrse problem.
or by pllolothermal defleclion1W-ll2
392 Infrared and Thermal Testing
Depth Profiling in Frequency with the reconstruction depth Zreo
Regime fulfilling the condition:
In this approach the specimen is with Drcc a constant. Thus Eq. 39 is
transformed:
rilluminated by a wide pump laser beam
modulated at frequency to generate a
plane thermal wave for investigation of
rinternal thermal properties. In particular
frequency drives the penetration depth J
of the thermal waves:
I~(38) I D (42) Snmm(r}-1 -J'o dln[e(z,,)j
Snonn{f) + 1
' •f 2dzrec
0
where Dis thermal diffusivity (mz.s-1). At x exp[-2(1 +i)z,,
high frequencies the induced thermal JX 1;:, ]dz,"
waves have a short penetration and may
interrogate the surface thermal properties; It is worth noting that any inversion
at low frequencies, the thermal waves procedure from Eq. 42 may only
have a high penetration and may reconstruct the thermal effusivity depth
interrogate deeper layers. Obviously the profile e(Zrcd as a function of Zren which
whole thermal depth profile may be unfortunately differs from z. However in
reconstructed by considering the many applications the heat capacity may
photothermal signal in the whole be assumed to be constant to estab1ish a
frequency range. According to thermal link between effusivity, conductivity and
wave backscattering theory126· 128 a clear diffusivity depth profiles. In such a case,
direct relationship between the effusivity once e(zrcc) is calculated from Eq. 42, the
profile e(z) and the photothermal signal function z(Znx·) may be obtained:
frequency spectrum S(f) is found. The
basic formula for thermal wave single
scattering may be expressed for thermal
reflectivity as follows:
(39) Snorm(f)-1 r(r) (43)
Snorm{f)+ 1
Consequently combining e(Zrecl with
-J~ dln[e(z)j z(zrec) the real effusivity profile e(z) is
determined.
2dz
The main problem is now the inversion
0 of the integral in Eq. 42. If the thermal
reflectivity for N different frequencies is
x exp[-2(1 + i)N measured and the thermal effusivity
profile in the number L of reconstruction
x J do Jdz depths Zrcc,i is divided, then the integral in
Eq. 42 may be replaced by the summation
o jD(o) and finally reduced to a linear system.
where d is the total derivative, j is an The problem of reconstruction nmv
imaginary unit, Snorm is the normalized
consists of solving such an m posed
photothermal signal, z is depth (meter), &
is depth (meter) and r is thermal system of 2N equations (one set for the
real and another set for the imaginary
reflectivity (dimensionless): part of r) in L unknown quantities with
the help of the singular value
(40) Snmm (f) s(r) esu.r decomposition (SVD) mathematical tool.
Sref{r) eref Singular value decomposition analyzes the
matrix L x 2N in terms of eigenvectors
where S is the signal of a heterogeneous and relative eigenvalues and uses just a
sample under test, Src:r is the signal of a few of them for the inversion. In fact the
reference homogeneous sample measured lower eigenvalues generally lead to a clear
instability in the reconstruction. The
in the same conditions, e5urf is the surface criterion of selection consists of the
effusivity of the heterogeneous sample definition of a threshold eigenvalue ),111
under test and t'ref is the surface effusivity and is given by using for the inversion
of the reference homogeneous sample only eigenvalues larger than this
threshold. On one hand this procedure
measured in the same conditions. permits reconstruction of stable profiles;
The integral in Eq. 39 may be
simplified by replacing the real depth z
Data Processing and Modeling for Infrared and Thermal Testing 393
on the other hand it limits the spatial spectrum has been calculated and
resolution in reconstruction, which different levels of gaussian percentual
noise have been added. For such a system,
strongly depends on the choice of ). 111•
As an example the reconstruction the L = 70 calculated eigenvalues decrease,
almost logarithmically, from 0.09 to
procedure is described for the particular 2.4 X I0-17 (0.090, 0.Q45, 0.024, 0.014, ...).
conductivity profile shown in Fig. 32
(continuous line). 131 The heat capacity is For the reconstruction, shown in
assumed constant so that conductivity
Fig. 32, only a number A of eigenvalues
and effusivity are proportional. The values more than a certain threshold ),111 is used.
The four charts refer to the different noise
L = 70, N = 200 and Zmax = 2 mm (0.08
in.) are fixed whereas the frequencies are levels in the signal: (a) 0.1 percent,
(b) 0.5 percent, (c) 1 percent,
logarithmically spaced between 1 Hz and
2.5 kHz. In the figure the frequency (d) 5 percent. In each chart the
FIGURE 32. Thermal wave backscattering used to reconstruct conductivity profile (W·m-l.K-1) versus depth for different
numbers of threshold eigenvalue A. Four different gaussian noise percentage levels: (a) root mean square noise = 0.1 percent;
(b) root mean square noise= 0.5 percent; (c) root mean square noise= 1 percent; (d) root mean square noise=- 5.0 percent.
Dimensions of linear system are L =- 70, 2N = 400. 1 W-m-1-K-1 = 7 BTU1c·in.·h-1.ft-2·°F-1.
(a) (c)
60 60
~ ~
~E 50 ~E 50
;t ;t
w 40 w 40
e ~
"-
"'"- B"' 30
-c0o
0 u0 20
., 30
•<J
u
-0co
u0 20
0 0.5 1 1.5 2.0 0 0.5 1 1.5 2.0
(0.02) (0.04) (0.06) (0.08) (0.02) (0.04) (0.06) (0.08)
Depth, mm (in.) Depth, mm (in.)
(b) (d)
60 60
.~, .~, 50
E 50 E
;t ;t
"'w 40 w 40
~ ~
"-
:"e' 30
i? 30
-c0o
·B
u0 20 -c0o
20
0 u0
0.5 1 1.5 2.0 0 0.5 1 1.5 2.0
(0.02) (O.D4) (0.06) (0.08) (0.02) (0.04) (0.06) (0.08)
Depth, mm (in.) Depth, mm (in.)
legend
D = conductivity k (VII.m-1.K-1) for threshold eigenvalue A = 2
+=conductivity k (W·m-1·K-1) for threshold eigenvalue A = 5
<>=conductivity k (W·m-1·K-1) for threshold eigenvalue A = 7
1\ =conductivity k (W·m· 1.K-1) for threshold eigenvalue A= 10
X = conductivity k (W·m-1.K-1) for threshold eigenvalue A = 14
394 Infrared and Thermal Testing
reconstructed conductivity is shown as a Experimental Results of Depth
function of depth for different choices of Profiling in Frequency Regime
the threshold A1h or, equivalently, A (A = 2, Some experimental results on hardened
5, 7, 10, 14). As predicted, the increasing steel materials serve as a first example of
number of eigenvalues has two the depth profiling in frequency regime.
The hardening process is a thermal cycle:
counterbalancing consequences: it (1) a heating process to reach the
complete auste11ization of the steel and
enhances the spatial resolution but (2) a very fast cooling to obtain the
increases the risk of instability, especially martemitic structure. The martensitic
·when noise increases. The best structure exhibits a higher hardening
property. In many industrial applications
reconstruction comes from a tradeoff the hardening process makes it possible to
increase the hardening of the steel,
between stability and spatial resolution.
These two requirements individuate an FIGURE 33. Thermal wave backscattering reconstruction errors
as function of threshold eigenvalue ],1h: (a) root mean square
optimum value for A. signal error; (b) root mean square conductivity error.
Dimensions of linear system are L = 70, 2N := 400.
A more quantitative treatment may be
done on the basis of error analysis. (a)
Actually once the profile is reconstructed, 1.0
two different kinds of a posteriori errors
may be considered: the error in fitting the ~
signal S(f) and the error in fitting the m
profile k(z). Obviously, for noiseless "Sf 0.1
signals, they are strongly correlated,: if the
c
first error tends to zero, the second error
tends to zero too, for the uniqueness of m
the solution. Something different happens ~
for noisy signals where a perfect fit in the
signal space generally does not correspond E 0.01
to a perfect fit in the profile space.131,132 To
0e
understand how to enhance the
reconstruction quality, it is important to gc'
0.001
study the behavior of both errors as a
~
function of the threshold eigenvalue Ath·
In Fig. 33 the two root mean square "'c
errors (root mean square signal error and "'Vi 0.0001
root mean square conductit1il}' error) are
plotted as a function of A1h in the w-t lQ-3 1Q-S 10-7
conditions of Fig. 32. The different
Singular value decomposition eigenvalues
symbols refer to different noise levels in (arbitrary unit)
the signal. As A,h increases, the root mean (b)
square signal error generally decreases but,
beyond an optinmm value Aopi1 the error 10
stops decreasing and reaches a constant
~9
level corresponding to the signal noise m
level. 8
S"f
On the other side, as A1h increases, even cm_ 7
the root mean square conductivity error E~ 6
~.
decreases, reaches a minimum value just 5
eo-;
for A1h = Aopt and then starts increasing. E
The mathematical reason of such behavior gc'-:;-';- 4
~3
is that by increasing Ath it is possible to
improve the quality of the signal fit till ~
the root mean square error becomes of the "'e 2
order of the noise. Beyond this critical
~
point corresponding to Ath = Aopt any
0 1Q-3 lQ-S 10 7
effort to fit better the signal is useless; 1Q-1
moreover the singular value
Singular value decomposition eigenvalues
decomposition procedure tries to get (arbitrary unit)
information even by fitting the noise and, legend
as a consequence, the relative
T = 5 percent r10ise
conductivity profiles become unrealistic X = l percent noise
and unstable. A. = 0.5 percent noise
• = 0.1 percent noisE'
In conclusion it is always possible to + = 0.01 percent noise
0 = noiseless
find an optimum reconstruction
condition by looking at the root mean
square signal error and by working out the
optimum tlueshold eigenvalue Aopt· Such
a value is reduced by the noise that acts as
a loss of spatial resolution over the
reconstructed profile.
Data Processing and Modeling for Infrared and Thermal Testing 395
transforming into martensite the surface Photothermal radiometric signals ·were
layers up to a suitable depth L in the measured for three samples:
millimeter range (hardening depth),
depending on the applications. Because 1. One hardened sted sample is
martensite has lower thermal conductivity thermally heterogeneous.
than austenite, the hardened steels arc
macroscopically thermally heterogeneous 2. The same hardened steel sample was
examined after excision of a 140 pm
and may be described by a thermal (S.S x Io-3 ill.) thick surface layer. In
conductivity depth profile k(z) practice it may be considered as a new
heterogeneous sample.
corresponding to the in-depth hardening
3. One homogeneous steel sample is used
process. as a reference.
Concerning the other thermal
The photothermal radiometric signals
parameters, the heat capacity has no of both steel samples (cases 1 and 2)
significant changes from austenite to should be normalized to that of the
martemite and therefore may be assumed reference sample (case 3)i the
constant: this means that both normalization is a standard step for the
inversion as described in Eq. 39 and
conductivity profiles and diffusivity moreover makes it possible to reduce the
profiles are proportional to each other. systematic errors in the measurement. The
normalized signals Snonn((J of case 1 (+)
FIGURE 34. Normalized photothermal radiometric signals and case 2 (0) are plotted versus the
versus frequency square root: (a) amplitude ratio; (b) phase frequency square root: the amplitude ratio
difference (degree). is in Fig. 34a and the phase contrast is in
Fig. 34b. It is worth noting that for both
(a) cases 1 and 2, the phase contrast is
positive, corresponding to the condition
1.1
ksurf < kuulk• as expected in any hardening
1.0 process. By using the singular value
decomposition procedure, the normalized
..0, signals in Fig. 34 may be inverted to
reconstruct the best diffusivity depth
e profiles for both cases 1 and 2 as plotted
in Fig. 35.
~
Because for both cases 1 and 2 the
"0 0.9 sample is the same, if the reconstructed
deptll profiles of 140 J.lm (5.5 x J0-:1 in.)
~ are shifted appropriately as in I:ig. 35, the
·Q". two profiles should superpose. The slight
E differences visible on the figure give
<(
0.8
•
0.7
0 4 8 12 16 20 24 28
Frequency square root, ''(Hz) FIGURE 35. Diffusivity depth profiles from radiometric data in
(b) Fig. 34. Reconstruction is performed by using thermal wave
backscattering theory and singular value decomposition
30 algorithm.
25 -c 0.20 (1.86)
-;;- E
1' 20
i'"' 0.15 (1.40)
"~'
. -k
~ 15
' ••
uc~ 10 • 1u 0.10 (0.93)
1'
~ • D.
it; 5 ;>
"0 :~ """"
~ 0 •D o " ' D +ooo 11 D ·~ 0.05 (0.47)
0
~
~ it;
"" <llo
~ """' "'D "0
'l.o ~ D ho
-5 D •D
D "E 0
~
D ~
-10 4 8 12 16 20 24 28 1- 0 0.5 1 1.5
0 (0.02) (0.04) (0.06)
Frequency square root, "(Hz) Depth, mm (in.)
legend legend
-1- = hardened steel sample, thermally
+ = hardened steel sample, thermally heterogeneous heterogeneous
0 = same hardened steel sample after excision of 140 prn 0 = same hardened steel sample after excision of
140 1-1m (5.5 x 10·3 in.) thick surface layer
(5.5 x 10-3 in.) th"1ck surface layer
- :o- theoretical best fit
396 Infrared and Thermal Testing
quantitative information of the error 100 Hz). The different slopes correspond
about the procedure. As a further to the change of the average thermal
validation note in Fig. 34 the good quality diffusivity, ·which decreases with x, from
of the fit between the experimental data D = 0.25 cm 2·s-l to D = 0.08 cm 2.s-l, as if
(symbols) and the continuous curves the process could inhibit the heat
corresponding to the profiles in Fig. 35. In diffusion. The distortions from linearity
summary, it may be observed that the are due to different diffusivity depth
hardening process corresponds to a profiles D(z), which have been
change of diffusivity from reconstructed in Fig. 37 according to
Duu!k = 0.2 cm2·s-1 to thermal wave backscattering theory. The
D5urf = 0.08 cmz.s-1.127 asymmetric behavior could be explained
by considering that in the experimental
As a second example the depth cell, the primary current distribution was
profiling on a thin palladium layer asymrnetric.12s
170 pm (6.7 x J0-3 in.) thick helps to
reconstruct the thermal diffusivity during Depth Profiling Using Lateral Scan
the electrochemical loading that forces
hydrogen into the sample. In general such In this case a pump laser beam,
a loading process generates both a high modulated at a suitable fixed frequency f,
hydrogen concentration and a high is focused on the sample to generate a
concentration gradient at the surface, spherical thermal wave useful to
which creates a stress field.l2H This investigate the internal thermal
phenomenon generates discontinuities properties. Even in this case the spherical
and dislocations that inhibit the heat thermal wave propagates inside the
conduction and diffusion. As a result the sample and is backscattered by the
thermal diffusivity should decrease by thermal heterogeneities. However the
increasing the hydrogen concentration. backscattering phenomenon occurs not
The photothermal radiometric only in the z direction as for plane waves
measurements have been performed on but also in oblique directions. ln fact the
the same sample during the loading spherical thermal wave may be seen as a
process, together with the standard superposition of oblique plane thermal
measurement of the electrical resistance to waves that propagate in various directions
control the loading ratio x (deuterium
palladium nucleus ratio). with angle 8 with respect to the z axis.
In Fig. 36 the amplitude of the thermal V\1hen these plane waves find the
reflectivity is reported in logarithmic scale scattering center at a depth z, they are
versus ~(f) for five different measured partially reflected back depending on both
loading ratios. All curves are straight lines effusivity and conductivity profiles (see
in a wide range of frequencies (1 to discussion of thermal waves in the
chapter on heat transfer) and eventually
FIGURE 36. Thermal reflectivity r versus frequency square root FIGURE 37. Thermal diffusivity depth O(z) profiles
reconstructed by using thermal wave backscattering theory
~(f) calculated from photothermal radiometric data on and singular value decomposition algorithm on data in
170 ~m (6.7 x 1Q-3 in.) thick palladium layer subjected to Fig. 36.
electrolysis.
E 1.0 ~-------------~
c 1: 0.30 (2.79) , - - - - - - - - - - - - - - - - ,
~
c 0.5 ~ Pure palladium
-.e~ vc
~ 0.3 T.. o.2o (1.86)
.e"'0 "cEv'. + •• +
0.2
'>
a.
~ 0.10 {0.93)
E
~
•">t 0.1 "" '0
"'jij ." . "§
"' 0.05 ~0
0 4 8 12 16 20 24 28
F. 0 50 100 150
Frequency square root, .V(Hz) (0.002) (0.004) (0.006)
legend Depth, J-Jm (in.)
0 = 0.02 measuring load ratio legend
+ = 0.30 measuring load ratio
+ = diffusivity D(z) at loading ratio 0.30
+ = 0.42 measuring load rat'lo 0 = diffusivity D(z) at loading ratio 0.42
II. = diffusivity D(z) at loading ratio 0.51
.6. = 0.51 measuring load ratio - = diffusivity 0{7) of unloaded samp!e
X = 0.60 measuring load ratio
Data Processing and Modeling for Infrared and Thermal Testing 397
reach the surface at the position any direction, according to the value of
x == 2ztan(8). In conclusion the surface hankel domain spatial frequency s. Note
temperature in any point x contains the that in the one-dimensional case, the
information on both effusivity and frequency doinain depth profiling
conductivity profiles. This enrichment of
analyzes the surface temperature by
the information content is due to the changing the imaginary part roof ~2 in
spherical wave instead of plane wave as in Eq. 45, driving the penetration of the
the frequency regime depth profiling. thermal \'•:aves. In the three-dimensional
case ~2 may be varied even by changing
To quantify the correlation between wave inclinations. By analogy the lateral
photothermal signal and the thermal
scan depth profiling analyzes the
depth profiles, the thermal wave temperature spatial spectrum U (K·m2)
backscattering model is considered for the verSUJ s and consequently temperature
spherical waves induced by a gaussian field T (kelvin) versus r, at constant
pump laser beam (three-dimensional heat
frequency to reconstruct two independent
diffusion).'" With the help of hankel thermal depth profiles. According to the
transformation a relationship may be thermal wave backscnttering theory the
~stablished between the temperature field surface temperature solution of Eq. 45
T(z,r) (kelvin) and its spatial spectrum
may be always put in terms of the thermal
0(z,s) (K"m2):
reflectivity r:
(44) ii(z,s) JJ,(sr)f(z,r)rdr (46) ii(2 = o,s) P e x-p( s-a-) 2-
8
0
kPz=O
Equation 44 is substituted into the heat + l'(z = O,s)
diffusion equation: X - r(z = O,s)
(45) p2 ii -d2-U + -I d-k d-U where Pis pump beam power (watt) and f'
dz2 k dz dz
is surface thermal reflectivity
2) ·[ Dim(z) + s U (dimensionless): 129
The three-dimensional heat diffusion is (47) r(o,s) _ ~J -azlanz(kP] exp(-zzp)dz
not restricted to the z axis but occurs in
0
FIGURE 38. Numerical example of reconstruction of thermal diffusivity and heat capacity depth
profiles.
0.20 (1.86) 4 (60)
E"ci 0.19 (1.77) 0 ------ oooooooo ~
0
'~ 0 DDDDDDD ~
0
-~ 0.18 (1.67) 0 i___[ :.e.0..
0 13 (45)
1 0 2.0
0 (0.08) ~
v 0
-+-0 y
"':~ 0.17 (1.58) 0
0 2 (30) "'·o
0 0 1 (15)
ro
."",0'. -~--.L._ 0vro.
§ 0.16 (1.49) 1.0 "v
(0.04)
v ~
.~... v
:Evv
oo
0.15 (1.40) 0.
0 ~
Depth, mm (in.)
legend
o = singular value decomposition reconstruction
- = true profile
398 Infrared and Thermal Testing
Equations 46 and 47 show a direct means over dz. Once the analytical
relationshjp between temperature spatial expression for temperature spatial
spectrum U (kelvin) and the profiles of
quantity k(z)p(z). Equation 4 7 has a clear spectrum 0 is obtained it is easy ·to
physical meaning: the internal changes of
calculate the surface temperature by using
kP act as backscattering centers for the the inverse of Eq. 44 and the lateral
·component of the photothermal
forward thermal wave. The total thermal deflection angle: 129
reflectivity at the surface may be obtained
by integrating the backscattered waves (48) <!>(x) ~ 4>,,(x)
over all the scattering centers- that
j+ cJIf[{ex+ (s~f _zzp
FIGURE 39. Photothermal deflection signal amplitude versus
x sin(sx)}
lateral offset for steel samples where frequency f = 1 Hz.
1000
> 7 [k(o)f+ ~(~) ]J
.3w,
.~ x d[lnkP] sdsdz]
100 dz
Q_
E
~
"'c
V"i'
1.0 2.0 3.0 4.0 where x is the lateral offset, <jl11 is the
(0.04) (0.08) {0.12) {0.16) deflection by the homogeneous sample
used as a reference and Cis a constant.
lateral offset, mm (in.)
The retrieval procedure to reconstruct
legend
both k and p (that means k, D and pc)
0 =hardened steel sample
+ = unhardened steel sample entails inverting Eq. 48, bringing z in a
grid and solving a linear system by the
means of sinsular value decomposition
(SVD) technique. Figure 38 shows
FIGURE 40. Thermal diffusivity depth D(z) profile reconstructed by singular value
decomposition at different frequencies (left scale). Also, micro hardness profiles from vickers
measurements (right scale).
0.17 (1.58) 1000
0
'c 0.16 (1.49) 0 00 J A¢6 ~ ft A«"A ~ fit A4A
(1.40) 0
E (1.30)
(1.21) +A '>i,. BOO
o-c=i- 0.15 (1.12) 600
(1.02) 0 oil Ao 400
1, .. ••• •" •
zoo
i 0.14 • 0
v
Ji. 0.13 • • •
:~ • • +
.", 0.12 + +• D
••
""'~.c 0.11
D
f- D0 0 0
0.10 (0.93) 0.5 1.0
0 (0.02) {0.04)
Depth, mm (in.)
legend
D = vickers hardness measurement
0 = diffusivity reconstruction at 4 Hz
+ "' diflusivity reconstruction Dt 9 Hz
A= diffusivity reconstruction at 16Hz
Data Processing and Modeling for Infrared and Thermal Testing 399
inversion results for both diffusivity and sample effusivity, Q is the energy
heat capacity and compares reconstructed
profiles with original ones. deposited per unit area and r is the
Experimental Results of surface thermal reflectivity:
Depth Profiling Using
Lateral Scan (50) r(o,p) f {~ -d ln[c(z)j
Zdz
The experimental results from hardened 0
(0) and unhardened (+) steel samples are
presented in Fig. 39 by using the ~xexp[-z.JPJo-'/D(o) ldz}
plwtothermal deflection technique. The
amplitude of the lateral component of the In case of weak thermal depth profiles, lrl
photothermal signal is measured as a << 1 and it may be assumed:
function of the distance from the heating
(51) T(o,p) ~ Qr;; [1 + zr(o,p)j
source. The frequency is fixed to f = 1 Hz.
e(oJ- p
Because of the near surface
modification of the microcrystalline Q {~-J~ dln[e(z)j
structure due to martensite formation, the
lateral scan from the hardened specimen e(o)[P 0 dz
deviates remarkably from the scan from
the unhardened. The deviations in xexJlzfp"Jo v~D-(ol)dz}
amplitude and phase carry all necessary
information to retrieve the two unknown Equation 51 may be antitransformed
independent depth profiles D and pc. The giving rise to an analytical expression for
feasibility as ·well as performance of the the temperature rise T(t). Usually this
lateral scmmins technique is verified by temperature is normalized to a reference
testing of surface hardened steel temperature 1~lt) of a homogeneous
samples with typical hardness depths of sample. In this case the surface
about 0.5 mm (0.02 in.). temperature ratio or any other
photothermal signal ratio may be written:
The mirase technique has been applied
for different frequencies~ I, 4, 9 and s(t)
16 Hz- at a scan length of about 4 nun
(0.16 in.). The thermal diffusivity (see (52) S,e~(t)
Fig. 40) and heat capacity profiles (see
Fig. 41) reconstructed for different
frequencies are in good correspondence
·with vickers microhardness measurements
from the sample's cross section. Note that
the profiles reconstructed from lower
frequency data present better accuracy
and spatial resohttion.t29
Depth Profiling in Time Domain where, as in Eq. 42, an equivalence for
Zr~'((z) has been stipulated,
Because a clear relationship exists between
frequency and time domain, when a Equation 52 represents the starting
heterogeneous specimen is illuminated by
a single-pulsed pump laser beam, the point of the retrieval problem to
cooling dynamic of the surface
temperature should contain the same reconstruct e(z). One simple procedure is
useful information to" reconstruct the given hy dividing the sample in a number
depth profiles k(z) and D(z}, as the of sublayers with constant effusivity to
frequency measurements described above. transform the integral into a summation
Even in this case, an extension of the of an algebraic line(lr system,
thermal wave backscattering theory may unfortunately ill posed as in a frequency
be introduced in the laplace domain, so
that the time laplace transform of the domain, which may be inverted by using
surface temperature may be ·written: no singular value decomposition.
(49) T(O 1>) - ___(].~ I + r(O,p)
' - c(o).JP 1 - r(o,p)
where p is the variable in the time laplace
transform domain, e(O) is the surface
400 Infrared and Thermal Testing
FIGURE 41. Heat capacity depth profile normalized to bulk FtGURE 42. Surface temperature versus time after ideal
value and reconstructed by singular value decomposition at heating pulse. Different curves refer to diffusivity profiles in
different frequencies. Fig. 43.
210 (18) ~-----------·-
0
1.0
Unhardened steel
e·0p
·B' 00
~ 0.5 ~ 1 (1.8) D
100
Q.
mv
1i B
~
g_
m
E
~
I
3 c
0 0.5 ' 1.0 ~
0 (0.02) (0.04)
.gv
0 0.1 (0.18) L_..L__L_J_L_i__L~ ___L_L_i__L__L_
V')
0 10
Depth, mm (in.) Time (ms)
legend
A step hardened steel sample
B graded hardened steel sample
c homogeneous sample
D another homogeneous sample
fiGURE 43. Thermal diffusivity depth profiles.
o.2o (1.86) c
0 -------~---=:==----~::==-----=-====----~==-=------·;;,=*~--~=-
: oo~o
B
0 0.5 1.0 1.5
0 (0.02) (0.04) (0.06)
Depth, mm (in.)
legend
A = step hardened steel sample
B = graded hardened steel sample
C = homogeneous sample at D = 0.20 cm 3-s· 1
D = homogeneous sample at D = 0.10 cm2 -s ·1
Data Processing and Modeling for Infrared and Thermal Testing 401
For example numerical simulations are
reported on a steel sample. The heat
capacity is assumed to be constant so that
only one indept:ndent lhermal parameter
exists, say, thermal diffusivity. The
dynamic of the temperature rise at the
surface is plotted in Fig. 42 for the four
different diffusivity depth profiles of
Fig. 43, which describe possible surface
hardening processes.
The differences in the temperature
values among homogeneous samples,
graded hardened steel and step hardened
steel are clearly seen already 100 ms after
the pulse. By using the singular value
decomposition procedure to invert the
data in Fig. 42 for the graded hardened
steel, a reconstructed depth profile
(Fig. 43) is obtained that only slightly
differs from the original depth profile. Of
course the quality of the reconstruction
profile is limited by noise in the surface
temperature dynamic as in other cases.
Conclusion
It is useful to compare thermal wave
backscattering depth profiling procedures.
Frequency and time domain depth
profiling have practically the same
performances because of the fourier
known relationship between time and
frequency. The frequency and time
profiles may reconstruct only one thermal
depth profile with the same accuracy and
sensitivity to noise but experimentally
there is a great difference: the time scan
takes several seconds whereas the
frequency scan may take at least several
minutes but with more measurement
accuracy.
On the other hand, lateral scan depth
profiling makes it possible to reconstruct
two independent thermal depth profiles
but with less accuracy and higher
sensitivity to noise.
402 Infrared and Thermal Testing
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of Physics (1999): p 37-39.
410 Infrared and Thermal Testing
.. .
CHAPTER
Thermal Contrasts in
Pulsed Infrared
Thermography
Jean-Claude Krapez, French National Aerospace
Research Establishment (ONERA), Chatillon, France
PART 1. Background to Thermal Contrasts •an
Pulsed Infrared Thermography
Thermal nondestructive testing has may thus reveal internal discontinuities.
developed to a powerful method for
discontinuity detection in various By comparing the therrnograms (that is,
materials, from metallic to composite the time versus temperature curves)
structures.
recorded at each suspected pixel to the
Different kinds of discontinuities are one corresponding to a sound area 1 it is
prone to reveal detectable contrasts in possible to get information on
thermographic images. A first class of
discontinuities leads to a modification of discontinuity parameters like depth, size
the normal functioning of the considered and even thermal resistance.
part in such a way that heat flow at the
surface is substantially affected. This There is a long history of research on
results in either a temperature decrease or actiw thermal nondestructive testing.
a temperature increase with respect to the
nominal level. The industrial fields where Many useful references before 1990 can be
such phenomena motivated the found in a review paper written by
implementation of infrared thermography Vavilov1 and in a few papers dedicated to
are the electric power production, electric discontinuity characterization.2•3 A
power distribution and the building area compilation of abstracts on thermography
(thermal insulation control). Hot or cold of composites can also be referred to:1
spots on the infrared images could Later works focus on aspects like
indicate loose or corroded connections, multidimensional modeling (either
undersized conductors1 fused circuits, numericaP.S·S or analyticai9-12), the
moisture ingress in walls or improper
insulation applications. These analysis of the respective influence of
discontinuities are generally observed in a
passive way1 that is, without any operator noise or pulse duration on the
intervention on the considered thermogram resultsi3-1S and the specific
equipment or material in process and in
particular without any supplementmy heat problem of corrosion detection and
itttroduction. characterization.l6
This chapter will actually deal with a Conditions for
second class of discontinuities. This class Discontinuity Detection
consists of discontinuities, separations1
structural or morphological modifications Several conditions must be satisfied for
- that is1 all kinds of material or structure efficient thermal nondestructive testing.
perturbations -that do not lead to any
significant modification of heat flow or 1. The sought discontinuity must
temperature in the part's service life. A significantly modify the additional
thermal impulse needs to be brought to heat flO'iN introduced by the operator.
the part for the thermal nondestructive
testing to be successful (Fig. 1). As FIGURE 1. Model for detection of resistive discontinuity at
opposed to the previous control process depth 11• Pulsed heating source thermally excites thick
the present thermal test is usually called structure. Infrared camera is located on same side as heating
active.
source.
Active Thermography
Pulsed heating source Discontinuity-
Heat is absorbed at a selected boundary of lhermal resistance of
the part and it is expected that the hidden / infinite lateral extent
discontinuity will disrupt the heat flow.
As a consequence of this disruption the r-~r-
temperature field is modified not only in
the core of the part but also on its Infrared camera ~
boundaries. An infrared camera directed Controlled structure
to one of these boundaries could record - infinitely thick wall
the temperature history of the field of
interest for a given period. Abnormal
contrasts during the thermal evolution
412 Infrared and Thermal Testing
2. Reciprocally the heat input through them is well approximated hy
characteristics must be adapted to the introducing a thermal resistance boundary
considered discontinuity, especially to condition.2,J,IU,ll The thermal resistance is
its geometry. the proportionality coefficient between
the thermal flux through the boundary
3. The temperature variations induced by and the temperature difference across it.
the discontinuity are of course not the
same on the different boundaries. The In steady state the thermal resistance n
infrared camera should therefore be
pointed to the surface that presents (K·m2-W--4) of a layer of thickness J and
the highest contrast.10,17 conductivity k (K·m2·VV-1) is given by:
Unfortunately this is not always (1) R ~ !__
possible: the front face, that is, the
one receiving the heat impulse, is k
sometimes the only one accessible for
temperature recording. This expression is also valid in the
transient regime corresponding to a
These three aspects are now developed thermal nondestructive test provided that
a little more. the time scale is large enough compared
to the diffusion time t (second) through
Cracks, delaminations, corroded parts the discontinuity:
(the corrosion product itself or the
material loss due to corrosion), high . ,z
porosity, disbands, water and oil ingress,
accidental implants -all these (2) t = -
discontinuities can be represented by a a
volume having modified physical
properties. In a particular region of the where a is the discontinuity thermal
inspected part a volume with altered diffusivity (m2 -s-1):
properties thus replaces the nominal
material. V~'hich properties must present a (3) a = -k
variation for a thermal contrast to appear pC
on the surface?
where Cis contrast O·kg-1-K-1) and pis
Here only the important case is material density (kg·m-:"~). It is common to
considered where the heat pulse is simulate cracks, disbands and
absorbed at the surface of the host delaminations merely though their static
material and where the material is opaque thermal resistance.
in the infrared spectrum of the camera.
The case is not considered •where heat is A discontinuity with a high thermal
deposited inside the material, as for resistance will hinder the heat diffusion to
example when a semitransparent material the bulk and the thermal contrast will be
is illuminated with a flash lamp or with a correspondingly higher. It will be seen
below that the impact of a given thermal
laser or when a dielectric material is resistance on the surface contrast depends
on its depth.
heated with microwaves. It is necessary
that either the thermal conductivity k or A heat flow distribution prone to be
the thermal density pC present a variation blocked by the discontinuity should be
in the discontinuity area (where p is selected. In that instance uniform heating
density and C is specific heat capacity). of the surface is suitable for detection of
discontinuities parallel to the surface like
In some instances the unique delaminations, disbanding and spread
parameter that conditions the appearance corrosion. On the other hand, uneven
of a temperature contrast at the surface heating of the surface is necessary for the
level is the thermal effusivity b ~ ~(kpC). detection of vertical cracks. Concentrated
Indeed \Vith one-dimensional heat flow it heating and scanning of the surface can
is impossible to detect the interface he used. The flying spot system derived
between two materials having the same therefrom showed to be efficient for crack
effusivity. The surface temperature detection in metals and composites.zn.22
evolution is the same as for a unique
material.18•19 Most often the Test Protocol and
discontinuities present both a reduced Discontinuity Morphology
conductivity and a reduced thermal
density. As a consequence the effusivity The nondestructive testing t'onfiguration
has locally large variations. Indeed for an considered in this chapter is basically as
important class of discontinuities ~ follows: the surface of the part uniformly
among them cracks, disbands and absorbs a short thermal excitation and an
delaminations ~ the foreign material is infrared camera monitors its temperature
air. Such effusivity variations can lead to distribution.
high temperature contrasts at the front
surface. The experimental results obviously
vary with pulse duration. It was shown
Generally the discontinuities are so
thin that their heat capacity can be
neglected. In that case the heat flow
Thermal Contrasts in Pulsed Infrared Thermography 413
that the optimum heating protocol Below, a series of theoretical abacuses
depends on a variety of factors, especially highlights the influence of the main
the type of noise (additive or thermophysical parameters on the
moltiplicative). 13 Anyway it can be essential characteristics of the contrast
demonstrated that the visibilitF of a spatial distribution and time evolution.
discontinuity (expressed as the contrast The results are presented below in a
normalized by the energy input or by the general \Vay without sticking to a
running temperature over an anomaly particular material (nondimensional
free area) is the highest when the pulse parameters are therefore intensively used).
duration is infinitely short (dirac pulse). One of the advantages of this approach is
The theoretical results here are thus based that isotropic and anisotropic materials
on the hypothesis of a dirac pulse. In the can be considered simultaneously.
case of a fbtite duration pulse, a first order
correction consists in moving the time Thermal Parameters
scale origin to the pulse barycenter.23 The
thermogram can then be compared \Vith The absolute contrast Ca(t) is defined:
the one computed with the hypothesis of
a dirac pulse. (4) Ca(t) = T(t) - T,r(t)
The present discussion is confined to where T(t) is the temperature recorded
the case of discontinuities parallel to the over the anomalous region and TrcJ(t) is
front surface, specifically delaminaHons the reference temperature measured over a
and corrosion. Delaminations are discontinuity free area. This contrast is
simulated by a thin thermal resistance proportional to the absorbed energy
and corrosion is simulated by a cavity provided by the pulse thermal source.
open to the rear face. The discontinuities Some kind of normalization is thus
are either circular (a disk shaped needed for a generalization of the results.
delamination or a flat bottom hole) or
very elongated (a ribbonlike delamination The choice for this normalization
or a flat bottom groove). The host depends on the total thickness of the
material may be anisotropic. tested material. In the case of a
semiinfinite model (simulation of a verv
The purpose is to provide theoretical thick material) temperature can be ·
results about the thermal contrast induced normalized with the adiabatic
by such discontinuities. The main temperature level of the first la}'er, that is,
informative parameters are the contrast the one over the discontinuity (see Fig. 1):
maximum and the time occurrence of this
maximum. An interesting parameter is where Q is absorbed energy density, pis
also the time \Vhen the contrast reaches material density, C is specific heat and /1
half of its maximum. These two time is discontinuity depth.
values provide a time interval useful for
managing the thermographic recording. In the case of a finite thickness model,
The maximum value of the contrast is by it is preferable to normalize temperature
itself very important: the parametric with the adiabatic temperature level of
analysis defines the limits of the method the discontinuity free material:
(the discontinuity size limit, the depth
limit and consequently the resistance (6) Tr,ad _g__
limit beyond \Vhich no detection is
possible). It also provides a valuable tool pCI,
for sizing the excitation source (the
energy density input necessary for safe where total thickness /1 :::: / 1 + /2, where /2
detection of a particular discontinuity, is the thickness of the layer behind the
given the temperature resolution of the discontinuity (see Fig. 5, below).
infrared camera).
Depending on the simulated case the
All these questions are answered below normalized contrast C11(l) is then defined:
by introducing models progressively more
sophisticated (one-dimensional and then (7) c.,(t) c,(r)
hvo-dimensional models). This refinement
is particularly necessary because the or Tl,ad
physical process yielding the thermal
contrast is a diffusion process: the c.,(r) c~, (t)
appearance of the discontinuities is thus
far from their real shape. So the contrast Tt,ad
level and its distribution are in a complex
relationship \Vith the discontinuity's real
size and shape, with its depth, with its
proximity to the rear face and with its
resistance. Some formulas of limited
application have been suggested.I0,24-2o
414 Infrared and Thermal Testing
This time function has a maximum the contrast reaches half of its maximum
Cn,max (see Hg. 2). This value is reached at
time tn,ma.x- Because of the involved value.
physical process of heat diffusion this The normalized c;ontrast is convenient
time is expected to be of the order of
1?-a.z- 1 where a.z is the host material for sizing the pulse energy. It will
diffusivity in the through-thickness therefore be considered systematicaiJy
direction: below with some numerical examples.
(8) _k,_ Nevertheless another type of contrast is
pC sometimes considered in the literature:
where kz is conductivity in the the relative contrast1 also called the running
through-thickness direction. Hmvever a contrast:
large number of parameters (closeness of
the rear face, lateral dimensions of the (10) c,(t) T(t) - T,1(t)
discontinuity, thermal resistance etc.) T,d(t)
prevent the real trend from being exactly
Jike f12·az- 1• This motivates the Again, three characteristic parameters can
introduction of the fourier number be extracted: the maximum value Cr max of
Fon,max: the relative contrast1 the fourier nml1bt.r
For,max corresponding to the maximum
(9) Fon,max occurrence and the fourier number
For,max/2 corresponding to the time when
The fourier number Fon,max/2 is defined in the contrast reaches half of its maximum.
the same way from the time ln,max/2 when
It should finally be mentioned that
FIGURE 2. Typical curve of normalized contrast in pulsed throughout the presentation heat losses
thermography. Definition of characteristic parameters Cn,max• through convection or radiation are
ln,max and tn,max/2· Fourier numbers Fon,max and Fon,max/2 are disregarded. This approximation is valid
deduced from both time values by application of Eq. 9. when the biot number hl1·kz-1 is much
Same holds for relative contrast and for its three lower than 1 (h is the transfer coefficient
characteristic parameters. Present curve actually corresponds at the front surface and is typically
to case of circular flat bottom hole wi!_h relative depth
around 10 w.m-2·K-1 in usual conditions).
It ·ft-1 == 0.5 and normalized diameter d1 == 2. Time scale
This criterion is generally satisfied in the
already corresponds to fourier number scale. case of metals (high conductivity
materials). Jn the case of poor heat
0.30 conductors, if the discontinuity depth is
too high, then specific abacuses for which
heat losses were taken into account must
be consulted.
0.25
0.20
0.15
0.10
0.05
0 1.0
0.1
Time (arbitrary unit)
legend
Cn =normalized contrast (see Eq. 7)
Cnm~' ==maximum normalized contrast
Cn,;~x/1 == half of maximum normalized contrast
d1 ==normalized diameter (see Eq. 21)
11 == total thickness
11 == discontinuity depth
tn.=< =time when normalized contrast reaches its maximum value
tn.=~n =time when normalized contrast reaches half its maximum value
Thermal Contrasts in Pulsed Infrared Thermography 415
PART 2. One-Dimensional Model of laterally
Extended Discontinuity
Surface temperature evolution in the case a signal-to-noise ratio lower than 1. From
of a laterally extended resistive Fig. 3a or from Eq. 13 the detection
discontinuity can be calculated threshold is found:
analytically. Of the techniques that have
been proposed 2•10•27 the quadripole (14) k, > o.o55
technique 10 is easiest to implement.
This means that a discontinuity 'Whose
Extended Discontinuity in resistance is lower than about 5.5 percent
Semiinfinite Wall of the front layer resistance cannot be
detected. This is an absolute limit because
\•Vhen the discontinuity is simulated with an other parameters, as for example the
a thermal resistance Rat depth /1, it finite lateral size of the discontinuity and
becomes clear that the normalized and the proximity of the rear surface,
relative contrast depend on only two contribute to further reduce the relative
parameters: the fourier number azt·l(2 and contrast.
the normalized discontinuity resistance, that
is, the ratio beh-..'een discontinuity Now an example of pulse energy sizing
resistance and the resistance of the front may be considered. The problem is to
layer: define the excitation energy necessary for
the detection of a 20 ).UTI (8 x ]()-4 in.)
The absolute contrast is normalized with thick delamination, 2 mm (0.08 in.) deep
the adiabatic temperature level of the in a very thick carbon epoxy plate.
front layer T1,ad (see Eqs. Sand ?a).
The air conductivity is 0.026 \".'·m-1-K-1
The maximum values of both the and the composite conductivity is taken
normalized contrast Cn max and the to be 0.64 \.Y-m-l·K-1• The normalized
resistance of the considered delamination
relative contrast Cr,max depend only on R1•
is thus R1 = (0.02 + 0.026)/(2 + 0.64) =
They are rising functions (see Fig. 3). An
immediate consequence of this 0.25. This value is higher than the limit
specified in Eq. 14, so the discontinuity
dependence on R1 is that a discontinuity should be detected provided the energy
input is high enough. According to the
of a given resistance R will be easier to abacus in Fig. 3a and Eq. 12 the induced
detect when it is close to the front surface. normalized contrast theoretically reaches
a maximum value of 0.052.
For low values of normalized resistance
both contrasts rise proportionally to.R1• It is now assumed that the detection
Then their rise slmvs down. The following has to be made with an infrared camera
empirical laws are proposed. The precision whose noise equivalent temperature
is better than 3 and 4 percent respectively: difference is 0.05 K (0.05 'C = 0.09 'F) and
that the required signal-to-noise ratio is
It is generally admitted that natural 2:1. This means that the minimum
variations in the emissivity of black temperature difference between an
painted surfaces are about 2 percent. 17 anomalous area and a not anomalous area
This limit can be used to distinguish should be 0.1 K (0.1 'C = 0.18 'F). The
detectable from nondetectable adiabatic temperature T1,ad of the first
discontinuities. Indeed a resistive layer should thus be higher than
discontinuity inducing a surface contrast 0.1 + 0.052 = 1.9 K (1.9 'C = 3.4 'F). From
Cr,ma:-. of no more than 2 percent will give the value of 1.5 x 106 j-K- 1 -nr-~ for the
thermal density pC of graphite epoxy the
energy density value Q = 5700 j-m-2 is
inferred. A flash lamp or an infrared lamp
must be selected and laid out close
enough to the inspected surface so that
the energy density reaches at least this
level.
The fourier numbers Fon,m~x and
Fon,mJx/2 are plotted versus the normalized
resistance in Fig. 3h. The corresponding
characteristic times for the relative
416 Infrared and Thermal Testing
contrast For,mJx and For,max/Z are plotted in temperature field. This will give the time
necessary for thermographic recording. In
Fig. 3c.
the case of the previous delamination
For small values of R1 the four fourier (R1 = 0.25)1 the characteristic fourier
numbers Fon,max/Z and F011,m;~x are 0.32 and
numbers are nearly constant. This means 0.76. The diffusivily of carbon epoxy in
that in this case the time occurrence of the direction normal to the plies is
typically 4.3 1Q-7 m2-s-1• The so-caJJed
the maximum contrast and of the half rise diffusion time l/·a7- 1 through a thickness
/ 1 of 2 mm (0.08 in.) is then
is proportional t o /12·a7- 1. For higher 22 + 0.43 = 9.3 s. The normalized contrast
time occurrences also actually reaches its maximum value at
values of R11 the
depend on R1• fu,max = 0.76 X 9.3 = 7.1 s after the pulse. It
The abacuses in Figs. 3b and 3c can be
used to evaluate the time necessary for an
internal discontinuity to build a
detectable print on the surface
FIGURE 3. Effect of discontinuity normalized resistance 7?1 = Rk7 ·11-1: (a) on maximum relative contrast Cr,max and on maximum
normalized contrast Cn,max; (b) on fourier numbers corresponding to normalized contrast curve; (c) on fourier numbers
corresponding to characteristic points of relative contrast curve.
(a) (c)
10.0
1.0
0.1
O.Ql
0.001 j !! 1.0 10.0 100.0
0.01
0.1
Normalized resistance ii1 Normalized resistance 'R1
(b)
3:;-----== ..10.0
-~~~j
-- -~__,,;
1--,-
.ll
E
c"
0,01 0.1 1.0 10.0 100.0
Normalized resistance ii1
l~gend
Cn "' normalized contrast (see Eq. 7)
C,"""' "' maximum value of normalized contrast
· C, = relative contrast (see Eq. 10)
C,,ma. = maximum value of relative contrast
Fo,.,_,.,,~, == fourier number corresponding to maximum normalized contrast
Fo,.,,m~,11 = fourier number corresponding to half of maximum normalized contrast
Fo,,ma.. = fourier number corresponding to max·tmurn relative contrast
Fo,, ,11 ,Jl = fourier number corresponding to half of maximum relative contrast
k, = conductivity in through-thickness direction
R = resistance
R1 = normalized discontinuity resistance (ratio between discontinuity resistance and resistance of front layer); R·kA -l
Thermal Contrasts in Pulsed Infrared Thermography 417
reaches half of its maximum at Cr,milx and from previous computation
tn,max/2 = 0.32 X 9.3 = 3 s after the pulse.
of /1•
As far as the relative contrast is
concerned the corresponding tirne The value of Fon.max/2 is recommended
occurrences are found by referring to instead of Fon,max for two reasons. n1e
Fig. 3c: tr,max ::::1.2 X 9.3 = 11.2 sand first one is that the experimental
lr,max/2 = 0.42 X 9.3 = 3.9 S.
determination of the abscissa of a
F.ach fourier number is uniquely related maximum is less precise than the one of
to the relative contrast maximum value the point at half rise. The second one is
Cr,max· It is possible to take advantage of related to lateral diffusion problems.
this relation for inversion, that is, to
evaluate the depth of a detected Indeed the objective is to apply these
discontinuity. Two particular relations are inversion formula on real discontinuities,
reported in Fig. 4. Best polynomial fits are
reported below with accuracy better than that is, discontinuities with a finite lateral
±3 percent: extension. It was shown that lateral
diffusion around the discontinuity only
(15) 1-'or,max 0.988 + 1.95 cr,max
progressively affects the contrast
+ 0.868 C~max evolution. Therefore the experimental
(16) For,max/2 0.386 + 0.397 Cr,max value of lr,max/2 is less perturbed by lateral
diffusion than is that of tr,max·
It is possible to consider even earlier
data: in the emerging contrast teclmiqtiC; the
discontinuity depth is inferred from the
time value when contrast actually emerges
from noise.25·28
+ 0.0853 cfmax Extended Discontinuity in Plate of
Finite Thickness
From the experimental measurement of
the maximum relative contrast Cr,max over As mentioned in the definition of the
a discontinuity the associated fourier thermal parameters, it is nnw convenient
to normalize the absolute contrast with
number For,max or For,max/2 is first the adiabatic temperature level of the
computed according to Eq. 15 or Eq. 16, whole plate (see Eqs. 6 and 7):
respectively. Then from the measurement
of the corresponding time occurrence (17) Cnl [T(t)-T,r(t)j pC/1
lr,max or lr,maxl2 the discontinuity depth 11 Q
is obtained through the fourier number
definition in Eq. 9. Equation 13 can be The llC'\V parameter in the present
used later to infer the discontinuity analysis is the relative depth / 1 ·11~1 of the
resistance R from the measurement of discontinuity (Fig. 5). It actually appears
that a discontinuity close to the rear face
FIGURE 4. Correlation between maximum relative contrast FIGURE 5. Model of resistive discontinuity in plate at relative
Cr,max and two characteristic fourier numbers. This correlation depth /1·/t-1. Rear surface is so close to discontinuity that its
can be used for discontinuity depth identification. presence must be taken into account.
Pulsed heating source
1~__, --,Discontinuity
-thermal
resistance of
infinite lateral
extent
Maximum relative contrast C,,m~~ Infrared camera
legend legend
C,,ma~ = maximum relative contrast (see Eq. 10) It = total thickness
Fo,ma, = fourier number corresponding to maximum relative contrast
11 = discontinuity depth
Fo,,~a•ll = fourier number at half of maximum relative contrast
11 _, thickness of layer behind (or under) discontinuity
418 Infrared and Thermal Testing
is more difficult to detect, that is, ·when deeper discontinuity is seen with the same
11·/t-1 comes close to 1. This difficulty
normalized contrast provided its
could be qualitatively explained as due to
resistance is proportionally higher1 that is,
the fact that the thermal echo coming from provided the discontiriuity keeps the same
the discontinuity merges with the one
coming from the back surface. Thermal ih value (see Fig. 3a). This was actually for
testing relies on a diffusion process, not a
propagation process like ultrasound. The the contrast normalized by Q·(pC/1)·1• It
echoes are thus far more spread in time. means that the absolute contrast was
This makes it difficult to separate the two
echoes previously mentioned. The inversely proportional to the depth /1• In
induced contrast is consequently very low. a finite thickness wall1 when the
discontinuity comes close to the back
This appears on the abacus of the
maximum contrast Cn,max in Fig. 6a. It is surface the absolute contrast drop is even
recalled that in a semiinfinite wall a
more acute. In Fig. 6a the contrast drops
by nearly two_orders of magnitude when,
for the same R1 value, a discontinuity
moves from a relative depth of 0.1 to 0.9.
fiGURE 6. Effect of discontinuity normalized resistance R1 = Rkz·/1- 1: (a) on maximum normalized contrast Cn,max; (b) on fourier
number Fon,max corresponding to maximum normalized contrast; (c) on fourier number Fon,max/2 at half of maximum
normalized contrast; (d) on relative contrast Cr,max· Different values of its relative depth {11-/1- 1 = 0.1, 0.2, ... 0.8, 0.9 and 0.95
from top to bottom).
(a) (c)
10.0 P:==;.. .... ' r~ II
'
~ 1.0
tf '
"c~ 0.1 c~!Z '
u0 :::::-·
v"'"0 0.01
·""§ ' '
z0 0.001 ~- -1
i j, '.
Ii
,,I_;_ L 1 I · - ' .J i~
·~. ;,,
0.0001 ' 0.1
i 0,01
O.Ql 0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0
Normalized resistance R1 Normalized resistance R1
(b) (d)
'10.0 0.0
f .. . :L:, I I l. -· ; .
" .• 1 ' ~: f.-+'.!' ':
1 I •.. 1.0
t "'~rI I I~ ·---,,! 'I '; I
I ' y I
II I ' I f- ' ' '
:;; 1:11 !::: '' '
''
.D , Ii !. , 1: . I 0.1
'
E 1.0 c--·-
~ .,., I ~ i 0.01 •' '
c I i i ~- '-L~±= I' •'
D:;;
' .,,·c
~
0
I I 'I!! I I i
~
I I'. . i.
, II ! , I; I
0.1 ! ' 11 'I . I',1 : ! ! I
O.Ql 0.1 1.0 10.0 100.0
Normalized resistance R1 Normalized resistance R1
legend
Cn.m~< = maximum normalized contrast (se~ Eq. 17)
C,,m" =- maximum relative contrast (see Eq. 10)
Fo,,m.l.< = fourier number corresponding to maximum normalized contrast
Fo,.,,,.12 = fourier number corresponding to half of maximum normalized contrast
k, = conductivity in through-thickness direction
R = resistance
R1 = normalized discontinuity resistance (ratio between discontinuity resistance and resistance of front layer)= R·k,·/1- 1
Thermal Contrasts in Pulsed Infrared Thermography 419
This is actually an important drawback for The same is valid for the maximum
value of the relative contrast (see Fig. 6d).
thermal nondestructive testing: The criteria for a safe detection expressed
discontinuities close to the rear face art' in Eq. 14 now depends on the relative
very difficult to detect in the reflectiull or depth of the discontinuity. J:or a
oue-sided configuration. The trammission discontinuity at relative depth O.S, 0.6,
or two-sided configuration is sometimes 0. 7, 0.8 and 0.9 the threshold value in
more suited but the access to the rear face Eq. 14 is respectively changed:
is not always possible. (18) R1 > 0.063, 0.080, 0.12,
0.24 and 0.91
The proximity of the back surface
contributes to reduce the characteristic \'Vhen the ratio between the discontinuity
resistance and the front layer resistance is
fourier numbers Fon,max and Fon,max/2 as
illustrated in Figs. 6b and 6c. This
reduction is particularly important for
high values of the normalized
resistance R1.
~URE 7. Effect of relative depth /1·ft-1 of discontinuity: (a) on maximum normalized contrast Co max; (b) on fourier number
'n,max corresponding to maximum normalized contrast; (c) on fourier number Fon,max/2 corresPonding Whalf of normalized
tntrast; (d) on relative contrast Cr,max· Discontinuity resistance normalized by resistance of whole plate, R1 :::::::: Rkz·/1-1, is 0.01,
03, 0.1, 0.3, 1, 3, 10, 30 from bottom to top.
1) (c)
100.0 1.0
~ 10.0 ~
f
o'
•;( 0
2 1.0 ~
5v
~
0 .0
~ 0.1 E
~c
"~z O.Ql
·~
0.001 ~
0
i'
(b)
0.8 LO 0.1 ·- l 0.4 ~~- LO
100.0 0 0.2
0.6 0.8
Relative depth /1./1- 1
(d)
10.0
~ 1.0
u"
t;(
g
c
u0 0.1
~
·~
""' 0.01
0.1 L_ 'c
0 0.2 0.4 0.6 0.8 0.001 0.2 0.4 0.6 0.8 1.0
Relative depth lr/1- 1 1.0 0 Relative depth 11·11- 1
legend
Cn.ma~ = maximum normalized contrast (see Eq. 17)
C,.m.~ = maximum relative contrast (see Eq. 10)
Fon,ma, = fourier number corresponding to maximum normalized contrast
Fon.max/2 = fourier number corresponding to half of maximum normalized contrast
kz = conductivity in through-thickness direction
R = resistance
ii1 = normalized discontinuity resistance (ratio between discontinuity resistance and total thickness resistance)= RkA-1
420 Infrared and Thermal Testing
higher than this threshold the generated applied to /12-az-1 to get the two
relative contrast is higher than the characteristic times ln,max and tn,max/2·
conventionflllimit of 2 percent.
Figure 7d is the i).bacus describing the
In the previous example of Cl
delamination 2 mm (0.08 in.) deep maximum value of the relative contrast
(R\ :::: 0.25), it can immediately be versus relative depth. Selecting a
threshold contrast value (0.02 is
concluded that detection will be possible commonly chosen) immediately yields
only if the plate in which it is located is
thicker than 2 7 0.8 :::: 2.5 mm (0.1 in.). the relative depth down to \Vhich a
discontinuity of given absolute resistance
Let it be assumed that previous
delamination is in a 3.3 mm plate can be detected.
(relative depth is 2 + 3.3 = 0.6). According
to the abacus in Fig. 6a the induced
normalized contrast theoretically reaches
a maximum value of 0.072. VVith the
same hypothesis regarding minimum
detectable temperature as before, it is
inferred that the minimum energy density
is now Q = 6900 ;.m-2.
Because the proximity of the rear face
leads to contrast reduction, more energy
is necessary to detect the considered
discontinuity.
Figures. 6b and 6c can be used to
evaluate the time necessary for an internal
discontinuity to build a detectable print
on the surface temperature field. The
characteristic fourier numbers Fon,maxt2
and Fon,max corresponding to present
example are 0.30 and 0.58. The
normalized contrast reaches its maximum
value at tn,max = 0.58 X 9.3 = 5.4 s after the
pulse. It reaches half of its maximum at
tn,max/2 = 0.30 X 9.3 = 2.8 s after the pulse.
The proximity of the rear face makes
the discontinuity disappear much faster
than if the material was very thick.
An alternative way to present the
contrast characteristic parameters is to
plot them versus the discontinuity relative
depth by using as a secondary parameter
the ratio between the discontinuity
resistance and the resistance of the entire
plate:
(19) 11,
The corresponding abacuses more clearly
reveal what happens as a discontinuity of
a given absolute resistance is moved into
the plate. Figure ?a shuws how rapidly
normalized contrast drops as the
discontinuity goes deeper. The decrease is
particularly important near the two faces.
For the lowest considered resistance
values, the drop is like (lt'T1- 1) 2 near the
front face and like (/2·It-1)2 near the rear
face. The decrease is less steep for higher
resistance values.
Figures 7b and 7c illustrate the
evolution of the fourier numbers Fon,m~x
and Fon,ma:</2 when a discontinuity with
given absolute resistance goes deeper and
deeper into the plate. The two abacuses
can be see11 as an illustration of the
multiplicative factor that should be
Thermal Contrasts ·m Pulsed Infrared Thermography 421
Part 3. Two-Dimensional Model of Discontinuity
with limited Lateral Extension
Previous abacuses dealt with the case of shaped discontinuity can then be applied
very extended resistive discontinuities. to the original complex shaped
Heat deposited by the pulsed source discontinuity ·with a good confidence.
penetrated down to the discontinuity. Indeed heat diffusion and its strong
Then because of the large extension of the tendency to blur all the discontinuity
discontinuity l1eat had no choice hut to geometrical details speak in favor of this
diffuse across it. In the presence of a procedure.
discontinuity with limited extent, heat
can diffuse laterally and bypass it. Two shapes vl'ill actually be considered
Previously there was heat accumulation here: the disk shape (circular flat bottom
above the discontinuity; now heat can be holes, circular delaminations) and the
evacuated around it. The direct strip shape (flat bottom grooves,
consequence is that the temperature ribbonlike delaminations). The synunetry
contrast on the front surface will be of the model is respectively axial and
unavoidably lower. This contrast planar. This leads to a geometry reduction
attenuation will be all the more important from three-dimensional to
as the lateral size of the discontinuity is two-dimensional. Computation savings
reduced. The purpose is here to provide are considerable.
abacuses that describe this feature.
The interaction between close
Two types of discontinuity will be discontinuities sometimes must be
considered because of their great considered, depending on the ratio of
importance in nondestructive testing: flat their mutual proximity to their depth.9
bottom holes and thermally resistive The close proximity of elements like
interfaces. Flat bottom holes serve as stiffeners can also have an influence on
models for discontinuities generated by the observed contrast. However, the
corrosion, whereas thermally resistive presentation below is restricted to the
interfaces are used as a model for isolated discontinuities in pJane plates of
delaminations and disbanding. uniform thickness.
In the real life these discontinuities can Definition of New Parameters
have of course any shape: the contours
are irregular and the discontinuities can One additional geometrical parameter is
even be multidomain (indeed in here introduced: d, the lateral size of the
mechanically impacted composites, discontinuity {diameter or \'\'idth,
multiple delaminations can be found, depending on the considered geometry).
stacked one over another). The simulation Many theoretical and experimental works
of the thermal response of a discontinuity have illustrated the importance of the
with a given complex shape is of course ratio of a discontinuity's size to its depth.
feasible but must be repeated for each This ratio has a direct consequence on the
new kind of shape. Finally a general trend discontinuity visibility by thermography.
will be difficult to extract from these Indeed visibility increases with this ratio.
computations.
However for the same ratio value the
Another approach consists of discontinuity vi.sibility changes from
simulating the thermal response of a isotropic to anisotropic materials. It often
series of discontinuities with a simpler happens that in anisotropic materials the
shape, like a disk or a strip. Their shape is in-plane conductivity is larger than the
then characterized through a unique through-thickness conductivity (for
parameter, namely thrir diameter or their example, in glass epoxy but even more in
width. If one had in hand the thermal carbon epoxy). Lateral diffusion is
response of a large panel of such basic heightened and the visibility worsens.
discontinuities, one would then have a Delarninations or disbondings in such
good idea of the response of a more anisotropic materials therefore need to be
complex shaped discontinuity. Indeed much wider than in isotropic materials for
from the aspect ratio of such a the same nondestructive testing success.
discontinuity (rather round or elongated)
and from the value of its smallest width, It appears that the discontinuity
the most representative disk shaped or analysis can be performed sitmtltmleously
strip shaped discontinuity can be selected. for both isotropic and anisotropic
The results pertaining to this simple materials if the size~to-depth ratio is
considered from the point of view of heat
diffusion. Not merely the geometrical
422 Infrared and Thermal Testing
features {Width d and depth 11} but also observed. Furthermore the smoothing
the corresponding diffusion times should
be compared. The governing parameter is intensity increases as il1 decreases. To
in fact:
solve this problem it was suggested to
where a., is the host material diffusivity in consider on the contrast images either the
the through·thickness direction and a_.. is locus at maximum gradient,3,5,29 the locus
at half maximum8•12 or the locus at
the in-pl<111e diffusivity. 40 percent of the contrast maximum.10
Depending on the symmetry of the The so-called apparent lateral $ize
determined by this way is close to but not
problem (axial or planar)/ al is called the equal to the real value. Furthermore. the
result depends on whether the considered
normalized diameter or the nonualized contrast image is an early one or a late
width of the discontinuity. one (it has been noticed that early images
provide more reliable .results/'~· 12·~8 des~ite
Thanks to normalized variables the the reduction of the stgnal to notse rat1o).
A quantified analysis of the correction
abacuses presented up to now apply tq_ that should be applied to the
experimentally measured apparent size to
any material. In the same spirit, with d11 get the real counterpart b presented
the abacuses presented below apply to below.
both isotropic and anisotropic materials.
Starting from noisy experimental data
In nondestructive testing one of the it is more precise to determine the locus
operator's tasks is to determine the actual of the contrast half maximum than the
locus of its maximum gradient
size of the discontinuity the operator (differentiation of an experimental signal
managed to detect. The drawback ·with leads to poor results unless appropriate
thNmal techniques is that the hot spot filtering is applied). Therefore the
present on the infrared images. is sel~on: a theoretical apparent lateral size considered
perfect representation of the dJscontmwty here is defined from the former one (see
true shape. For example in the case of a Fig. 8). The instant when it is measured
corresponds to the one when the
flat bottom circular hole1 the profile of normalized contrast reaches its maximum.
the infrared signal across the Even though the discontinuity
discontinuity has not the expected top representation has already deteriorated by
hat shape. A kind of smoothing is always this time1 the signal-to-noise ratio on the
contrast image is maximum.
FiGURE 8. Typical surface profile o~ normaliz~d co.ntrast. ~t
this time contrast over discontinuity cen.ter IS at 1ts max1mum Numerical Models of
value Cn,max· Discontinuity apparent size is defined from ful~ Thermal Contrasts
width at halt maximum. It is here slightly larger than real s1ze
d (12 percent overestimation). This profile.actuall~ The simulation of thermal transfer in a
corresponds to circular flat bottom hoje w1th relatJve depth material containing a discontinuity of
/1-/t-1 = 0.5 and normalized diameter d1 = 2. limited extent requires either an
approximated analytical approachY-12 or
0.3 a numerical approach.3·s-s,Js.t?,26
0.25 A two-dimensional finite difference
model was developed1 based on the control
0 volume approach and on the alternate
direction implicit procedure.3,5 The mesh
t; has a higher density near the front surface
and in the vicinity of the discontinuity.
g 0.2 By this means a fine spatial resolution of
c temperatures is achieved only in the area
0u where the thermal gradient is expected to
"0 0.15 be high. The time grid is nonuniform as
welL These characteristics allow a rapid
.& (only 60 x 60 cells are used) yet precise
"E 0.1 computation of temperature field
z0 evolution in the anomalous plate.
0.05
This discussion has focused on the
0 temperature evolution of the surface cell
-1.5 -1.0 -0.5 0 0.5 1.0 1.5 just over the discontinuity center. The
normalized contrast and the relative
Abscissa normalized by discontinuity diameter d contrast were derived from it as for the
infinitely extended discontinuity in a
legend finite thickness plate. The essential
C0 =normalized contrast (see E1_1. 17)
Cn,rnH =maximum value of normalized contrast
Cn rnax!l = half value of maximum normalized contrast
· d =diameter
d1 = normalized diameter (see Eq. 21)
/1 = total thickness
/1 =discontinuity depth
Thermal Contrasts in Pulsed Infrared Thermography 423
parameters are again: Cn,max1 Fon,max> Circular Flat Bottom Hole
Fon,max/2 and Cr,max· The temperature (Axial Symmetry)
surface profile that has developed at the
time corresponding to Fon,max {that is, the The geometry first analyzed is described
normalized contrast at its maximum value
Cn,rnaJ is used to define the apparent in Fig. 9. The flat bottom circular hole is
lateral size of the discontinuity (full width commonly used as a model for corrosion
at half maximum - see Fig. 8). The ratio simulations.
of this apparent size to the real one is
computed to provide a correction factor The finite difference computation
to be applied on experimental data.
results are summarized in Figs. 10 and 11.
The follmving series of discontinuity A series of 156 different configurations
parameters have been considered:
(I) relative depth 11·101; (2) normalized were taken into account for the
diameter or vl'idth, either by first layer
construction of these abacuses (different
thickness d1 or total plate thickness d1: combinations of relative depth and
(22) J, normalized diameter value).
In a first set of figures, the folluwing
and (3) eventually the normalized
a]are presented as functions of the
resistance {either by first layer thickness R1
or by total plate thickness R1): normalized diameter and for different
values of the relative depth: maximum
(24) R1
value of the normalized contrast Cn,max
As in previous computations, heat losses (Fig. lOa); corresponding fourier number
are ignored and it is assumed that the
excitation pulse has negligible duration. Fon,max (Fig. lOb); fourier number at half
FIGURE 9. Schematic of considered discontinuity: flat bottom rise of the normalized contrast Fon,max/2
circular hole. Model for corrosion simulation. {Fig. lOc); ratio between the apparent
diameter and the real diameter (Fig. 1Od);
X
maximum value of the relative contrast
Pulsed heating source Crmax (Fig. lOe).
Infrared camera ~ 'The plotted curves clearly illustrate
how the contrast characteristic parameters
legend I,
d"' discontinuity diameter evolve as the diameter of the
/1 =total thickness discontinuity is modified without
/1 ""' discontinuity depth
x"' direction normal to interrogating radiation changing its depth.
z "' direction of interrogating radiation
In particular it can be seen in Hg. lOa
that for holes of low diameter (that is,
d1 < 1 or 2) the contrast drops very fast
when the diameter is further reduced. The
trend is actua11y like d12. For the opposite
case, that is, for \Vide discontinuities,
there is a kind of leveling off. This is
particularly noticeable for discontinuitie~
at high relative depth.
The characteristic fourier numbers are
also rising functions of the normalized
diameter d1• However for discontinuities
of low diameter, they tend to an
asymptotic value that only slightly
depends on the relative depth of the
discontinuity (see Figs. lOb and !Oc).
Figure IOd shows that the apparent
diameter provides a slight
underestimation of the real diameter
value when d1 is larger than 2.5 (less than
15 percent error). For less extended holt's,
the full width at half maximum of the
contrast profile provides an
overestimation. This overestimation rises
very rapidly as i11 decreases. Small
discontinuities, assuming that they
provide enough contrast to be detected,
appear then much wider than they really
are.
As usual the Cr,max abacus (Fig. 10e) can
provide detection thresholds that are
based on particular hypotheses about the
irreducible experimental noise, namely
the emissivity erratic variations. By setting
the relative noise minimum level to
2 percent as above, the detection limit
424 Infrared and Thermal Testing
fiGURE 10. Effect of normalized diameter d1 of circular flat bottom hole: (a) on normalized contrast Cn,max (see Eq. 17); (b) on
fourier number Fon,m~x corresponding to maximum normalized contrast; (c) on fourier number Fon,max/2 corresponding to half
of maximum normalized contrast; (d) on ratio between apparent diameter and real diameter of discontinuity (arrows indicate
increasing relative depth); (e) on maximum relative contrast Crmax· Relative depth 11·/t-1 is 0.1, 0.3, 0.5, 0.7, 0.8 and 0.9 from
top to bottom. '
(a) (d)
10.0 r3.4 I
1 3.2
,J 1.0 B 3.o I
"b 2.8 '
c 2.4 ;
v0 0.1 2.2 ]--
2.0
'0
1.0 :·
.~ 0.8
"§ 0.01 0.1
z0
1.0 10.0 100.0
Normalized diameterd1 Normalized diameter il1
(b) (e)
10.0
1.0
0.1
~J_·_-r- .. ···~h11'
. ~+ ~- t fj·· 'Jli']001 ~-j- >i' · ·j ,
ji
0.001 _____;_ _ _I !._1I_:_L,Ii~-
· • .j 'ci. -'
·1',__t+j- . . ; -l
11, -~+- ~~~
;]~,:, _. :___ :_1~':_,1
0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0
Normalized diameterd1 Normalized diameter J1
(c)
1.0 !P T· . L Jl ~
N 1-- '
Ii
nvr- -trl
rH-:····x:;; I . I
, !n
0 :I
~
I~" I
c i!
"E' I
Fff iI,II."\'
' I;
L• i ,
0.1
d il j·~
I
l'u II 1
i
I 11:
i - j_llilil::
0.1 1.0 10.0 100.0
Normalized diameter il1
legend
Cn,m•x =maximum normalized contrast (see Eq. 17)
a,=C,""'K =maximum relative contrast (see Eq. 10)
normalized diameter (see Eq. 21)
Fo"·"'"~ =fourier number corresponding to maximum normalized contrast
Fon,m••ll = fouriN number corresponding to half of maximum normalized contrast
11 =total thickness
/1 =discontinuity depth
Thermal Contrasts in Pulsed Infrared Thermography 425
goes from (ft > 0.6 for shalluw maximum. Discontinuities are then
discontinuities (relative depth lower than measured through iterations.
O.S) to (11 > 1.2 for deep discontinuities
(relative depth of 0.9). First it can be noticed in Fig. tOe that
the mean value of Fon,max is about 0.3
Figures lOc and lOd can be jointly used (this value will be the starting point). The
for an inversion purpose. Let it be
assumed that a discontinuity of (nearly) value of tn,max/Z and the definition of the
circular shape was detected by pulsed fourier number (Eq. 9) permit a first
thermography. The purpose is then to estimation of the discontinuity depth /1 (if
evaluate its depth and its equivalent the through-thickness diffusivity is
diameter. Two experimental data are known).
needed: the discontinuity's apparent
On the other side the approximation is
diameter and time tn,ma:-.:12 ·when first made that the apparent diameter
normalized contrast reaches half of its really corresponds to the true diameter d.
FIGURE 11. Effect of relative depth /1·/t-1 of circular flat bottom hole: (a) on maximum normalized contrast Cn,max (Eq. 17);
(b) on fourier number Fon,max corresponding to maximum normalized contrast; (c) on fourier number Fon,m~x/l corresponding
to half of maximum normalized contrast; (d) on maximum relativ~ contrast Cr,max· Diameter normalized by total thickness is
0.25, 0.5, 1.0 and 2.0 from bottom to top, where total thickness dt = d-ft-1\l(u.z·ax-1).
(a) (c) =l:- I
100.0 1.0 ·--1 ·~
~ 10.0 a ·~··1--l
_____l__ I ~
0 0
'"g 1.0 0
c ~
u0 ~
.0
·""'""0 0.1
E
E
~
z0 0.01
c
·"c
~
.£
0.001 0.2 0.4 0.6 0.8 0.1 0.2 0.4 0.6 0.8 1.0
0 1.0 0 Relative depth /1'11- 1
Relative depth /1'/1-1
(b) (d)
100.0 10.0
~ <
c 1 1.0
~ 10.0
<.)'
".0
'"bc 0.1
E
u0
~
·"5
c ~
-~ 1.0 ""' 0.01
~
.£
0.1 0.0010 0.2 0.4 0.6 0.8 1.0
0 0.2 0.4 0.6 0.8 1.0 Relative depth /1·11- 1
Relative depth /1./1-1
legend
u:, = host materi~l diffusivity in through-thickness direction
«x == in-p!~ne diffusivity
Cn,ma. == maximum normalized contrast (see Eq. 17)
Cr,ma• = maximum relative contrast (see Eq.lO)
d = diameter
ii1 = normalized diameter (see Eq. 22)
Fon,m•' = fourier number corresponding to m~ximum normalized contrast
Fon,ma•/1 == fourier number corresponding to half of maximum normalized contrast
It = total thickness
426 Infrared and Thermal Testing
An approximation of i11 is then inferred of the normalized width d, (now
from the knowledge of the in·plane and normalized by the plate total thickness).
through-thickness diffusivity values
(Eq. 20). Next i/1 in Fig. lOd yields a ratio It is obvioqs that the contrast (both
between the apparent and real diameter
normalized and -relative) is higher with an
and permits a better estimation of the real
diameter. It is used for a ne\v calculation elongated discontinuity than with a
of il1 that serves as input for both circular one. The difference is however
Figs. lOc and 10d. Iterations can then more pronounced for discontinuities
proceed until convergence. having a low normalized lateral size
In a second set of figures, the following
(compare for example Figs. lOa and 13a).
are presented as functions of the
It can be noticed again that the
discontinuity relative depth and for
normalized contrast Cn,max rapidly drops
atdifferent values of the normalized
when eft is lower than a fe\\' units. The
diameter (that time it is normalized by
the total plate thickness): the maximum drop is less dra_Jnatic than \~th a circular
value of the normalized contrast Cn max
(Fig. 11 a); the corresponding fourief hole: it is like d1 instead of d 2 •
1
number Fon,max (Fig. 11 b); the fourier The fourier numbers related to the time
number at half rise of the normalized
occurrences of contrast maximum and
contrast Fon,max/2 (Fig. lie); and the
maximum value of the relative contrast contrast half rise are also larger for an
C,,max (Fig. lld). elongated discontinuity than for a circular
The plotted curves now illustrate how
one. Again the differences ar~ more
the contrast characteristic parameters
evolve when a discontinuity of given significant for low values of d1 (compare
first Fig. lOb with Fig. 13b and then
diameter is buried progressively deeper
into a plate. compare Fig. JOe with i'ig. 13c).
The results about the normalized The apparent ·width as observed on the
contrast and about the relative contrast
maximum contrast image provides a
(Figs. lla and lld) pinpoint the
tremendous attenuation encountered with slight !!nderestimation of the real width
thermal techniques. The normalized when c!J is larger than 4 (see Fig. 13d).
contrast attenuation is between 0.0006 for
small discontinuities (d1 = 0.25) and 0.007 \·Vhen d 1 decreases beluw 4, the
for wider discontinuities (d1 = 2) when the overestimation rapidly increases. In an
relative depth changes from 0.1 to 0.9.
This explains 'why it is so hard to detect isotropic material a_ groove whose width is
corrosion on the back surface at its onset, equal to its depth (a,~ 1) would actually
that is, when the discontinuity
appear twice as wide as it really is.
simultaneously is deep (as seen from the
front surface) yet has a poor lateral The Cr,max abacus (with Fig. 13c)
extension. indicates without any surprise that the
detection threshold is lowered in the case
of an elongated discontinuity. Indeed by
a]extrapolating the plotted curves, the
detection limit goes from > 0.12 for
shallow grooves (relative depth lower
than 0.3) to > 0.5 for deep grooves
(relative depth of 0.9). It is recalled that
fiGURE 12. Flat bottom groove for corrosion simulation.
Flat Bottom Groove (Plane Pulsed heating source X
Symmetry)
Infrared camera I,
The flat bottom groove (Fig. 12) can also d
be considered as a model for corrosion Legend
simulations. It describes the case of d =groove width '-t-t-~'
structured discontinuity, namely a very 11 = total thickness
elongated discontinuity. 11 =discontinuity depth
x =in-plane coordinate across discontinuity
The same series of 156 different
configurations as before was considered z '-'- depth coordinate
for the construction of the abacuses. The
geometry merely switches from cylindrical
to cartesian.
Figure 13 presents Cn,max' Fon,max•
Fon,max/l• the apparent width to real width
ratio and Cr,max as functions of the
normalized width , for six different values
of the groove relative depth.
Figure 14 presents Cn,max• Fon,mJx'
Fon,max/2 and Cr,max as functions of tht'
groove relative depth for different values
Thermal Contrasts in Pulsed Infrared Thermography 427
FIGURE 13. Effect of normalized width (}I of flat bottom groove: (a) on maximum normalized contrast Cn,max (see Eq. 17);
(b) on fourier number Fon,mJx corresponding to maximum normalized contrast; (c) on fourier number Fon,max/2 at half of
maximum normalized contrast; (d) on ratio between apparent width and real width of discontinuity (arrows indicate
increasing relative depth 11·11- 1); (e) on maximum relative contrast Cr,max· Relative depth /1·ft-1 is 0.1, 0.3, 0.5, 0.7, 0.8 and
0.9 from top to bottom.
(a) (d)
-5 4.0
1 ~
" 3.6
d
~
t:: 1 0 B
c~ -5
8 "0
~ '3
(ij 0.1 ~c
E ~
z0
~
0.01
0.1 Q_
Q_
~
0
·0g 1.0 10.0
_j_
"'1.0 10.0 100.0 0.1
Normalized width d1 Normalized width d1
(b) (e)
10.0 ! 10.0
1 J
t
~
c
~ 0v
~.0
E
" ·~·~c 0.1
c
"0 ;j-i
~
0.1 1.0 10.0 100.0 0.01
0.1 0.1
dNormalized width 1
(c) Normalized width d1
0.1 1.0 10.0 100.0
Normalized width d1
legend
Cn.m~'"' maximum normalized contrast (see Eq. 17)
Cr,max =maximum relative contrast (see Eq. 10)
d1 =normalized width (see Eq. 21)
Fo"·""' = fourier number corresponding to maximum normalized contrast
Fon.m~x/Z =fourier number corresponding to hill! of milximum normalized contrast
I,= total thickness
h = discontinuity depth
428 Infrared and Thermal Testing
shallow and deep refer to the distance from magnitude for high values of the relative
the discontinuity bottom to the front depth. The same holds for the relative
surface. contrast (compare Fig. lld with Fig. 14d).
By comparing Fig. lla with Fig. 14a it The characteristic fourier numbers are also
can be said that for the wider higher for an elongated discontinuity but
the difference is at most 52 percent for
discontinuity considered (i1t = 2) the
Fon,max and 26 percent for Fon,max/2
normalized contrast versus relative depth {compare Fig. llb with Fig. 14b on one
curve is nearly the same whether the
hand, Fig. 11 c with Fig. 14c on the other
discontinuity is circular or elongated. In hand). The maximum difference is again
the opposite case, for a smaller
observed for the smallest discontinuity
discontinuity (i11 = 0.25) the difference is
more significant. It can reach one order of (d, = 0.25).
FIGURE 14. Effect of relative depth h·l,-1 of flat bottom groove: (a) on maximum normalized contrast Cn,max; (b) on fourier
number Fon,max corresponding to maximum normalized contrast; (c) on fourier number Fon,max/l at_half of maximum
normalized contrast; (d) on maximum relative contrast Cr,max· Width normalized by total thickness, d1 = d·f1- 1·"\l(a.1·CJ.x-1), is
0.25, 0.5, 1.0, and 2.0 from bottom to top.
(a) (c)
100.0 1.0 ~
!
~ 10.0
~~'-
.J ;~ 1 -~ +- L +- __j
"'bc 1.0 II' I I.
0v _I L__!l II I
".~ 0.1 0.1 j_ - ___;_ -·· t
"E 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
Relative depth /1-/1-1
z0 0.01
0.001
0
Relative depth fl'lt-1
(b) (d)
100.0 _, 10.0
F~---s-
! 1.0
10.0 cf
0.1
--=-r--r-- ~· ---
1.0
.-.1 ·r:-_ .'0.1
~- i=ifl·· O.ot ~~--~~-~-~.oi=-=c=-.+i-- t~c:F~k~ ,_~
' _' j _ . I _ 0.001 .-~--+--~~ __ i -ere-:-=-t·---=r--=1t_-__-__j1'_, _-
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
Legend
ft, = host material diffusivity in through-thickness direction
H, = in-plane diffusivity
C,_m,; = maximum normalized contrast (see Eq. 17)
Cr,m~x = maximum relative contrast (see Eq. 10)
d =width
d1 = normalized \vidth (see Eq. 10)
Fon,max = fourier number corresponding to maximum normalized contrast
Fo,.,m~~u = fourier number corresponding to half of maximum normalized contrast
/1 = total thickness
11 = discontinuity depth
Thermal Contrasts in Pulsed Infrared Thermography 429
By applying the same procedure as for Previously there were only two
the circular holes, Hgs. 13c and 13d can adjustable parameters: (1) the relative
now he used for inversion. Through an
iterative technique the width and the aldepth lrlt-1 of the discontinuity and
depth of the discontinuity can be
similarly inferred after measuring the (2) either the width normalized by
ch<uacteristic time f11,max/2 and the full
width at half maximum of the contrast lateral size (Eq. 21) or the width (it
(apparent width). normalized by total plate thickness
(Eq. 22).
Disk Delamination (Axial
Symmetry) In the present case an additional
Previously the two-dimensional parameter appears, the normalized
discontinuity consisted in a hole. The
absence of matter forced heat to diffuse in discontinuity resistance R1 (Eq. 23).
the material nearby. In the case of a
delamination (Fig. 15) there are two Instead of using /1 for the normalization,
factors that reduce the perturbation to the the total thickness It yielding Rt can be
nominal heat flow induced. used (Eq. 24).
1. There is some material behind the Two-dimensional abacuses evolve
discontinuity, so heat can bypass the
discontinuity and diffuse into this therefore toward three-dimensional
volume. abacuses. For technical reasons such
2. Heat can be transferred through the representations are difficult to build and
delamination1 depending on its own to read as well. The analysis is therefore
thermal resistance.
restricted to one particular case for the
These two factors are at the origin of a thermal resistance. The scope will
contrast reduction. It is indeed more
difficult to detect a delamination or a unavoidably be more limited than with
disbanding than a simple hole. the flat bottom hole case, it will anyhow
provide useful information about the
There is thus a need for a new series of
abacuses. This would give tools to the expected contrast level and about the
nondestructive test operator for the maximum contrast occurrence.
choice of suitable thermographic
equipment and for the evaluation of the The follo-wing abacuses refer to the case
contrast generated by a given ofl~t = 0.5. They illustrate for example the
delamination or disbanding. case of a 40 pm (1.6 x I0-3 in.) thick air
FIGURE 15. Disklike resistive interface for simulation of circular layer in a 2 mm (0.08 in.) thick carbon
delamination or disbanding. epoxy plate or a 0.3 pm (1.2 x 10-5 in.) air
layer between two soldered aluminum
Pulsed heating source
alloy plates 2 mm (0.08 in.) thick each.
0 I, '-1--+Cd ...z Indications will be given later for an easy
Infrared camera extension of the results to lower R1 values.
Legend a]In a first set of figures, versus the
d "'0 discontirmity diameter normalized diameter and for different
11 "' total thickness values of the delamination relative depth
11 = discontinuity depth
x = radial coordinate are presented: the maximum normalized
z =- depth coordinate
contrast Cn,nwx (Fig. 16a); the
corresponding fourier number Fon,max
(Fig. 16b)i the fourier number at half rise
of the normalized contrast Fon,max/Z
(Hg. 16c); the ratio between the apparent
diameter and the real diameter (Fig. I6d);
the maximum relative contrast Cr,m;tx
(Fig. 16e).
By comparing the normalized contrast
generated by a delamination (Fig. 16a)
·with the one generated by a flat bottom
hole of same diameter and depth
(Fig. 10a)1 it can be noticed that the
former is much lower than the latter, as
predicted. The reduction is particularly
important for discontinuities with high
relative depth or low normalized
diameter,
I;or small delaminations the contrast
drop with i11 is steeper than (/12. At the
other end the leveling off when J1
increases is observed sooner than ·with
holes. The asymptotic limit corresponds
to the cll,!!l;i,\ value pertaining to the
one-dimensional model, see for example
Fig. 6a. By the way it can be noticed on
Fig. 6a that the normalized contrast Cn ma:..
is roughly proportional toRt when Rt is'
lower than about 1, This important
observation can be used to extrapolate the
430 Infrared and Thermal Testing
fiGURE 16. With disklike delamination, effect of discontinuity normalized diameter CJ1 (Eq. 20): (a) on maximum normalized
contrast Cn,max; (b) on fourier number Fon,max corresponding to maximum normalized contrast; (c) on fourier number ~on,max/2
corresponding to half of maximum normalized contrast; (d) on ratio between apparent diameter and real diameter of
discontinuity (arrows indicate increasing relative depth /1·/1-1); (e) on maxi11JUffi relative contrast Cr,max· Relative depth 1,·11- 1 is
0.1, 0.3, 0.5, 0.7, 0.8 and 0.9 from top to bottom. Normalized resistance Rt = Rk1·11- 1 is fixed to 0.5.
(a) 10.0 (d)
~
<f 1.0
"g
c
u0 0.1
·""E•"' 0.01
z0
10.0 100.0
Normalized diameter d,
(b) (e) 10.0
1 1.0
v'
0.1
0.01
0.001 1.0 10.0 100.0
0.1
Normalized diameter d,
Normalized diameter d1
(c) -~± t.-~
1.0 -. j-rc-.
+
~-~- ,·_+-'-r+H
+ 1.0 10.0 100.0
I
' Normalized diameter d,
0.1
0.1
legend k, "' conductivity in through-thickness direction
Cn.ma, "'maximum normalized contrast (see Eq. 17) 11 = total thickness
C.'"'' =maximum n~!ative contrast (see Eq. 10) /1 = discontinuity depth
R = resistance
d "'diameter
d1 = normalized diameter (see Eq. 24) R, = normalized d\scont\nu·1ty resistance (ratio between discontinuity
Fo,.,.0,, = fouriN number corresponding to maximum normalized contrast
Fon.ma•/Z =fourier number corresponding to half of maximum normalized resistance <Hld total thickness resistance) = R·k,·l, ·1
contrast
Thermal Contrasts in Pulsed Infrared Thermography 431
contmst results obtaint:>d in the present The ratio between the apparent
diameter and the real one follows the
analysis (R1 = O.S) to thC case of same trend as in the case of flat bottom
holes (compare Fig. 16d with Fig. 10d).
delaminations whose normalized There is an overestimation for small
resistance R, is lower than O•.S. alvalues of and then an underestimation
A leveling off is also observed on the a, a\for large values of (that is, when is
fourier number abacuses (Figs. 16b and larger than 2 or 3, depending on the
relative depth). However the
16c). In the case of holes, these numbers overestimation occurring for small
kept on rising with 'i11 (see Figs. 1Oh and
10c). In the case of delaminations, there is
an upper limit that depends on the
rt•lative depth of the discontinuity.
FIGURE 17. With disklike delamination, effect of discontinuity relative depth /1·ft-1: (a) on maximum normalized contrast Cn,max;
(b) on fourier number Fon,max corresponding to maximum normalized contrast Fon,max; (c) on fourier number Fon,max/2
corresponqjng to half of maximum normalized contrast; (d) on maximum relative contrast Cr,rnax· Di~meter normalized by total
thickness, d1 = d·ft-l.,l(az·U:..-1), is 0.25, 0.5, 1.0 and 2.0 from bottom to top. Normalized resistance, R1 = Rkz-Jt-1, is fixed to 0.5.
(a) (c)
10.0 1.0 ----I-i--!1-
~ ~ I '1---
.J 1.0 e _J 1
"g' 0 I i
c ~ I
u0 0.1
t '----
"0 .0
-
.,.~ E
c~
.~
•'C
Ez0 O.Ql
~
0
~
0.001 0.2 0.4 0.6 0.8 1.0 0.1 I 0.6 0.8 1.0
0 0
I_
0.2 0.4
(b) (d)
10.0 ·___- 1 10.0
:=+ I 1.0
i- -1----'
I 0.1
-- _l ±-
0.01
0.1 0.001 .
0 0.2
0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0
Relative depth /1-lt- 1
Legend
uL = host material diffusivity in through-thickness direction
ftx = in-plane diffusivity
Cn,max '-'- maximum normalized contrast (see Eq. 17)
C,m~x =maximum relative contrast (see Eq. 10)
' d = diameter
d1 = diameter normalized by total thickness (see Eq. 22)
Fon,~, =fourier number corresponding to maximum normalized contrast
Fon.~~x/Z =fourier number corresponding to half of maximum normalized contrast
k7 = conductivity in through-thickness direction
11 = total thickness
11 = discontinuity depth
R = resistance
R1 = normalized discontinuity resistance (ratio between discontinuity resistance and total thickness resistance)= R·k~·/1- 1
432 Infrared and Thermal Testing
discontinuities is smaller than with flat Then in a second set of figures (Fig. 20)
bottom holes.
Cn,maXI Fon,maxt Fon,max/2 and Cr,max are
The curves describing the relative presented as functions of the
contrast indicate that it will be atdelamination relative depth for different
experimentally more difficult to
values of the normalized width (it is
discriminate the delamination signature now normalized by the total plate
from noise as compared with the
thickness).
equivalent flat bottom hole (compare The asymptotic limits of the
Fig. IOe with Fig. 16e). By setting at
normalized contrast for large values of i11
2 percent the lower limit of the relative
are obviously the same as with disk
contrast for a safe detection, the following
shaped delamination (compare Fig. 16a
thresholds are found: d1 > 0.84 when the with Fig. 19a). They correspond to the
relative depth is 0.1; i11 > 1.6 when the
one-dimensional computation results (the
relative depth is 0.8. It is noticed that a discontinuity acts essentially through its
delamination at 0.9 relative depth cannot resistance, not through its lateral size). At
be detected, even if its diameter is very
the other end, the contrast drop \Yhen l11
large (the result is valid for R1 = 0.5 but
decreases below a few units is now less
according to Fig. 6d it can be extended to
dramatic.
resistances corresponding to R1 ~ 0.8).
The fourier numbers Fon max and
In a second set of figures (Fig. 17),
Fon,max/Z only slightly depe1ld on the
Cn,maXI Fon,uwx• Fon,max/2 and Cr,max are geometry (compare Fig. 16b with Fig. 19b
presented as functions of the
on one hand, Fig. 16c with Fig. 19c on
atdiscontinuity relative depth for different the other hand).
values of the normalized diameter The correction that must be applied to
(before the diameter is normalized by the the apparent width to get the real size is
total plate thickness).
similar to the correction pertaining to the
The plotted curves illustrate how the circular delamination case (compare
contrast characteristic parameters evolve
Fig. 16d with Fig. 19d). The
when a delamination of given diameter overestimation of the lateral size is now
and resistance is buried progressively
just a bit higher.
deeper into a plate. The relative contrast abacus shows that
The contrast attenuation when the
the constraint on the lateral size of the
delamination is progressively deeper
buried is even more acute than in the case discontinuity is now relaxed (compare
Fig. 16e with Fig. 19e). Indeed by setting
of a hole: the normalized contrast
attenuation when the relative depth is the lower limit of the relative contrast for
a safe detection to 2 percent, the
changed from 0.1 to 0.9 ranges between
following thresholds are found: d1 > 0.37
0.0001 for small discontinuities (rt, = 0.25) ·when relative depth is 0.1; i11 > 1 when
and 0.003 for wider discontinuities relative depth is 0.8. A delamination at
(l11 = 2). For this reason the deepest and/or 0.9 relative depth cannot be detected,
regardless of its width.
the smallest delaminations considered
cannot practically be detected. Indeed if fiGURE 18. Ribbonlike resistive interface for simulation of
the relative contrast threshold is set at elongated delamination or disbanding.
2 percent, it can be inferred from Fig. 17d
Pulsed heating source X
at : :that a delamination of normalized
diameter 2 cannot be detected if its
relative depth is larger than 0.85.
at : :Similarly a delamination of normalized
diameter 0.25 cannot be detected if
its relative depth is larger than 0.28.
Ribbonlike Delamination )' />' /•'' d
(Plane Symmetry) z
/
The same analysis is proposed for an //
elongated delamination (cartesian
geometry instead of cylindrical geometry). [/
The considered model is depicted in
Fig. 18. Infrared camera
The presentation of the abacuses is Legend
organized as before. A first set of figures
d = diswntinuity width
(Fig. 19) presents cll,lllJXt Cr,mJXt Fon,maxt 11 ""' total thickness
Fon,max/2 and the ratio of apparent width /1 -== discontinuity depth
to real ·width as functions of normalized x = in·plane coordinate across discontinuity
z = depth coo1dinate
width i11 for six different values of relative
depth of the ribbonlike delamination.
Thermal Contrasts in Pulsed Infrared Thermography 433
fiGURE 19. With ribbonlike delamination, effect of discontinuity normalized width (it (Eq. 20): (a) on maximum normalized
contrast Cn,max; (b) on fourier number Fon,max corresponding to maximum normalized contrast; (c) on fourier number Fon,max/l
corresponding to halt of maximum normalized contrast; (d) on ratio between apparent width and real width of discontinuity
(arrows indicate increasing relative depth= 11·11- 1); (e) on maximu.!,l1 relative contrast Cr,max· Relative depth is h·ft-1 is 0.1, 0.3,
0.5, 0.7, 0.8 and 0.9 from top to bottom. Normalized resistance R1 = Rkz·f1- 1 is fixed to 0.5.
(a) (d)
10.0 -.5, 2.8
1 -~ 2.6
..J 1.0 "1' 2.4
te; 8 2.2
~c L 2.0
.,0u 0.1 15
~ -~ 1.8 I
-~ ~c 1.6
"§ O.ot 1"
nn~.. 1.4
z0 ~
0 1.2
.g 1.0
~ 1.0 10.0 100.0
"' 0.8
0.1
Normalized width d1 Normalized width d1
(b) (e)
10.0
1.0
i ____r- ' -i 0.1
-i I',_ l1- I - 0.01
j1
:--+I I ': 0.001
' 0.1
Lilt illit tr0.1 __[ i
LI_~ J i U1 I'·
0.1 1.0 10.0 100.0 1.0 10.0 100.0
Normalized width d1 Normalized width d,
(c)
t~t' ~Vr-ttF-_
_ _j_
J" '
L_J_ cf-+-------:-:-:: '
'
I0.1 , IIiJl::I I, rll ! I jJii'
0.1 I! I jj
'- - '
1.0 10.0 100.0
Normalized width d1
legend Fo,,m., = fourier mnnber corresponding to maximum relative contrast
Cn,ma< = maximum normalized contrast (see Eq. 17) kz = conductivity in through-thickness direction
C,ma. = maximum relative contrast (~ee Eq. 10)
/1 = total thickness
d =width
=11 discontinuity depth
ii1 =- normalized width (see Eq. 21) =12 thickness of layer behind (or under) discontinuity
Fonm~' = fourier number corresponding to maximum normalized contrast R = resistance
Fon,,;3, 12 = fourier number corresponding to half of maximum normali1ed
R1 = normalized discontinuity resistance (ratio between discontinuity
contrast
resistance and total thkkness resistance)= R·kz-1.-1
434 Infrared and Thermal Testing
Conclusion thermal resistance). It is therefore
important to have a clear idea on the
Pulsed thermography can offer invaluable influence of these parameters. A
information on hidden discontinuities. quantitative analysis of the temperature
The physical variable monitored with this and of the contrast is necessary for several
technique is the material surface reasons: to sort between detectable and
temperature. But temperature varies in a nondetectable discontinuities, to properly
complex way with the discontinuity select the heat pulse energy level for
parameters (depth, size, shape and detection of a dass of discontinuities and
eventually to implement an inversion
FIGURE 20. Effect of relative depth 11·1t-1 of ribbonlike delamination: (a) on maximum normalized contrast Cn,max; (b) on fourier
number Fon,max corresponding to maximum normalized contrast; (c) on fourier number Fon,mJx/2 corresponding to half of
rraximum normalized contrast; (d) on maximum relative contrast Cr,max· Width normalize_9 by total thickness, with
d1 = d·/1- 1·--l(u.z·Ux-1), is 0.25, 0.5, 1.0 and 2.0 from bottom to top. Normalized resistance R1 = Rk1·11- 1 is fixed to 0.5.
(a) 0.2 ---~''---· ------- (c) 0.2 0.4 0.6 0.8 1.0
___ j____ _ Relative depth /l'/t-1 0.8 1.0
10.0 1.0
0.4 0.6 0.8 0.4 0.6
~ ~
..:! 1.0 1
"c~ r- 1
0v 0.1 ".0
.,."0 E
.§ "c
§ 0.01
·~
z0
,"\'
0.001'
0 --
0.1
1.0
(b) (d)
10.0 --=-0--- =cjc--= _j 10.0
i; l l- -J:_;\~ -1 ---+---+-+-+
_c::-j.c- ' I -~-_+--+- -+f--~- ~ 1.0
t--- ' __j_ ,_f
±1.0, \'~lt~t-;:::NI : t;
_[ b
•c 0.1
""'I !- ·50v ~----
~-
"•-" I~-.•T.
0.01
__ , - 0.001 0.2
0
0.8 1.0
legend
a, = host materiJ! diffusivity in through-thicknt'SS direction
a~ = in·p!Jne diffusivity
Cn,m~~ = maximum normalized contrast (see Eq. 17)
C.m~.' = maximum relative contrast (see Eq. 10)
d1 = discontinuity widlh normalized by total thickness (see Eq. 22)
Fon,m.- "" fourier number corresponding to maximum normalized contrast
Fon,m~-'ll = fourier number corresponding to hatf of maximum normalized contrast
k, = conductivity in through·thickness dir('Ction
11 = total thickness
!1 = discontinuity depth
R = resistance
fit "' normalized discontinuity resistance (ratio between discontinuity resistance and total thickness resistance)"' R·kA-1
Thermal Contrasts in Pulsed Infrared Thermography 435
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