![Figure 5: The respective intensity fringe patterns of Figure 4 normalised in the greyscale range [0, 1] (Equation 18)](/figures/figure-5-the-respective-intensity-fringe-patterns-of-figure-3u0lpltj.webp)
![Figure 6: The plate of Figure 4 when normalised in the greyscale range [0, 1] (Equation 18) and sampled at (A) 1, (B) 7, (C) 8 and (D) 6 kHz and a phase difference of 90](/figures/figure-6-the-plate-of-figure-4-when-normalised-in-the-yc90y6cs.webp)
On the Feasibility of Shearographic Imaging of
Acoustic Cross-Modulation
G. Kalogiannakis*
†
, B. Sarens*, D. Van Hemelrijck
†
and C. Glorieux*
*Laboratory of Acoustics and Thermal Physics, Dept. of Physics and Astronomy, Katholieke Universiteit Leuven, Leuven, Belgium
†
Dept. of Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Brussels, Belgium
ABSTRACT: The detectability of defects as well as their detection contrast have significantly
increased in recent years because of the use of nonlinear variants of the classical non-destructive
acoustic methods. The non-uniformity of stiffness and hysteresis around the area of a defect results
in strong non-classical acoustic nonlinearity, which can be used to effectively distinguish the presence
of defects. Motivated by the latest advances in nonlinear laser Doppler vibrometry and nonlinear
ultrasonics, we investigated the potential of shearography to detect the so-called cross-modulation
effect. This effect arises when a strong, low-frequency pump wave dynamically changes the elastic
response of a defect, so that the amplitude of a weak, high-frequency probing wave passing through
the defect is modulated. As a result, the frequency spectrum of the response contains mixed
frequency components, which are directly attributed to the underlying defect.
KEY WORDS: cross-modulation, non-classical acoustic nonlinearity, sandwich structures, shea-
rography
Introduction
Nonlinear variants of classical non-destructive
methods exhibit enhanced detection sensitivity for
defects [1–8]. Non-classical acoustic nonlinear phe-
nomena originate from specific boundary conditions
that are fulfilled near or at the interface of non-
bonded contact defects. Theoretical models typically
attribute this non-classical nonlinearity to the
asymmetry of stiffness in tension (opening) versus
compression (closing) of a defect as well as the hys-
teresis caused by friction or plastic deformation at its
borders. As a result, the asymmetric oscillation of
such an imperfect structure has an enriched spectral
content, the details of which depend on the partic-
ular type of excitation.
There are typically two different approaches to
investigate the presence of nonlinearity in a system.
First, one can excite the structure by a purely sinu-
soidal sound wave. A linear system in a dynamic
mode under harmonic excitation yields a sinusoidal
response at the same frequency. The spectrum of the
response of a nonlinear system or a locally nonlinear
system (such as a structure with a defect), on the
other hand, also contains higher harmonics of the
fundamental frequency [1, 2]. Depending on the
geometry of the structure, the depth and the size of
the defect, the higher harmonics are restricted to the
defect site so that their detection is defect-selective
with high contrast and good spatial resolution. Even
though this approach gives excellent results with
respect to detection contrast, it has the drawback
that it is vulnerable to the existence of inherent,
non-defect-related nonlinearities of some systems
(classical acoustic or electronic).
An alternative method that overcomes these
inherent defect-masking effects makes use of two
waves and is based on the so-called cross-modulation
effect. This effect arises when a strong, low-frequency
(X) pump wave modulates the amplitude of a weak,
high-frequency (x) probing wave through its influ-
ence on the dynamic response of the defect. As a
result, the frequency spectrum of the response con-
tains mixed frequency components (x · nX), which,
similarly to the previous method, are directly attrib-
uted to the underlying defect [3–8] (Figure 1).
Ballad et al. [3] describe a phenomenological model
where they consider the transmission of a small-
amplitude, high-frequency longitudinal or flexural
sound wave through a cracked defect area in a plate-
like sample subject to strong low-frequency vibra-
tions. The transmission coefficient depends on the
gap between the crack edges T(t) fi (T
0
+ T
(Xt))
causing linear modulation of the output response
V
out
(t) ¼ (T
0
+ T
(Xt)) · V
in
(xt), with a spectrum
containing mixed frequency components.
398 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd j Strain (2008) 44, 398–408
Zaitsev et al. [4, 5] reported another type of cross-
modulation based on the Luxemburg–Gorky (LG)
effect, where the pump itself is a modulated wave.
The modulation of the probing wave is attributed to
the influence of the crack opening to its dissipation,
which is also the case in conventional cross-modu-
lation. The difference lies in the fact that LG modu-
lation is not determined by the instantaneous
reaction of the material but rather the time-averaged
response over the pump carrier period. The dissipa-
tion is caused by strong thermoelastic losses at the
defect site. Despite the small size of the crack, these
losses are considered to be significantly increased
because of high-temperature gradients at the vicinity
of the crack edges, which are not determined by the
acoustic wavelength, but by the smaller scale of the
inter-edge microcontacts.
Hirsekorn [8] developed a theoretical model to
assess the quality of bond interfaces in composite
materials with compressional sound waves. Accord-
ing to this study, the reason behind the nonlinear
dynamic behaviour of adhesive bonds is the
increasingly nonlinear relationship of the binding
force when it approaches the bond strength. The
binding forces can be described by an interaction
force F(d), which depends nonlinearly on the dis-
tance d (opening of the interface from the average
static equilibrium distance d
e
), and can be described
by a polynomial expansion in a Taylor series as
follows:
Fðd
e
þ DdðtÞÞ ¼
X
1
n¼0
d
n
Fðd
e
Þ
dd
n
e
ðDdðtÞÞ
n
n!
(1)
The powers of the interface distance modulation in
Equation (1), when insonified by two compressional
waves at different frequencies x
1
and x
2
, are as fol-
lows:
ðDdðtÞÞ
1
¼ d
1
sinðx
1
t k
1y
yÞþd
2
sinðx
2
t k
2y
y þ uÞ
(2)
ðDdðtÞÞ
2
¼
d
2
1
2
f1 cosð2x
1
t 2k
1y
yÞg
þ
d
2
2
2
f1 cosð2x
2
t 2k
2y
y þ 2uÞg
þ d
1
d
2
fcos½ðx
1
x
2
Þt ðk
1y
k
2y
Þy u
cos½ðx
1
þ x
2
Þt ðk
1y
þ k
2y
Þy þ ug (3)
and so on for higher powers, where k
iy
are the wave
numbers of the different waves in the y-direction
(perpendicular to the bond) and u is the phase dif-
ference of the two waves. It is shown that the
dynamic response of the bond contains mixed fre-
quencies, which is the so-called cross-modulation.
Finally, in a recent article, Shkerdin and Glorieux
[9] demonstrate theoretically that cross-modulation
can also occur in plates with defects excited with
Lamb waves. In their study, they modelled the non-
linear interaction between a probing high-frequency
Lamb wave and a locally delaminated bilayer, whose
geometrical parameters periodically change because
of a low-frequency modulating Lamb wave. As a
result of this interaction, cross-modulation results in
the enrichment of the frequency spectrum of the
normal displacement signal evaluated at the free
Figure 1: Pump (200 Hz) and probe (2000 Hz) signals in terms of multiples of the optical wavelength, their cross-modulation and
its frequency spectrum
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd j Strain (2008) 44, 398–408 399
G. Kalogiannakis et al. : SHECROSSMOD
surfaces of the bilayer. Results from both non-atten-
uating and attenuating bilayers generally show a
significant localisation of the generated frequencies
around the delamination, and an improved delami-
nation detection contrast of the harmonics with
respect to the fundamental probing frequency.
Promising experimental techniques, which exploit
these defect-selective phenomena, have been devel-
oped in recent years based on laser Doppler vibrom-
etry [1] and ultrasonics [2–8]. The techniques that are
being used have the disadvantage that they operate
in a scanning mode. Depending on the desired res-
olution and size of the object under investigation,
such a test may be too time-consuming and therefore
cost-ineffective. This is the reason why full-field
methods like shearography [9–12], competitive
infrared (IR) thermography [13] and ultrasound IR
thermography [14, 15] are becoming increasingly
popular. Motivated by the latest advances in non-
linear laser Doppler vibrometry and nonlinear ultra-
sonics, we investigated the potential of shearography
to detect the so-called cross-modulation effect.
Speckle techniques like shearography typically
make use of a normal charge-coupled device (CCD
camera) to record an image of the object before and
after loading. They are based on the speckle effect,
which is a light interference phenomenon occurring
when a coherent light such as a divergent laser beam
is diffusely scattered on an optically rough surface
(roughness of the order of the optical wavelength)
[9–12]. The so-called objective speckles are forma-
tions of random optical interference in space from
light rays coming from different places on the object,
and their size and intensity change with the distance
from the object. Such a speckled image pattern of the
object under investigation is recorded using a camera
with each individual pixel acting as a separate optical
detector, which probes the displacement of a
respective area on the object. Adjusting the aperture
and the distance of the lens from the object controls
the size of the recorded so-called subjective speckle
and therefore the size of the area. Subtraction of
successive images before and after loading results in a
fringe pattern which is directly associated with the
change of the optical path and therefore the absolute
displacement [9] (or differential displacement [10–
12] for shearography in particular) of the object sur-
face under investigation.
In this study we analyse the influence of cross-
modulation and stroboscopic illumination on the
shape of the fringe patterns themselves. Another
approach could consist of obtaining stroboscopic
images at a very high temporal resolution. Each
fringe pattern obtained could then be demodulated
and integrated in order to obtain the absolute dis-
placement information. This way a spectral analysis
of the sample is straightforward and cross-modula-
tion effects should easily be observed. In practice
however, this approach has proven to be infeasible.
The demodulation process required manual input in
order to obtain reliable results, which made the entire
process very time consuming, while the result
remained disappointing.
In this study, we focused on the detection of del-
aminations between the skin and core of sandwich
structures. Sandwich materials are lightweight struc-
tures, which are especially oriented to satisfy design
requirements for high bending stiffness. To this end,
the core, a typically rather thick material of low
density and strength, is combined with thin (and
therefore low bending stiffness) skins of high tensile
strength [16]. It is evident that a critical factor for the
mechanical performance of the multilayer sandwich
structure is the quality of the adhesion between the
skins and the core. Manufacturing flaws and/or
compressive or impact loads, which may increase the
interlaminar shear stresses dramatically, may often
initiate the local debonding of the layers. Such
defects, also called delaminations, are very common
in sandwich structures. Their timely detection
becomes critical for the evaluation of the residual
strength of the structural component as this type of
defect may lead to local or global buckling and
eventually failure of the structure [17–20].
Theoretical Analysis
Before proceeding to the experimental investigation,
we present the layout of the theoretical model and
demonstrate the best approach to tackle the experi-
ment and data analysis. As an introduction to the
complicated nonlinear case, we first review the
method as it is used for linear vibrometry.
Stroboscopic dynamic shearography: linear case
When an object vibrates at frequency x, the optical
field scattered by a single point on the object, illu-
minated by laser light at time t, is given by the
expression:
U ¼ u
0
exp i/
0
þ i
4p
k
d
0
cos xt
(4)
where u
0
is the electric field oscillation amplitude, /
0
the phase at the view plane, k the optical wavelength
and d
0
the vibration amplitude.
400 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd j Strain (2008) 44, 398–408
SHECROSSMOD
: G. Kalogiannakis et al.
In shearography, the light U from a point is
superposed with the light U ¢ from an adjacent point
so that the intensity varies with time according to:
I / U
U þ U
0
U
0
þ U
U
0
þ
UU
0
¼ u
2
0
þ u
02
0
þ u
0
u
0
0
cos
4p
k
@d
0
@x
d
x
cos xt þð/
0
/
0
0
Þ
(5)
where we have considered, for simplification, that
shearing is only applied along the x-direction, with
the shearing distance being d
x
.
When stroboscopic illumination is applied at the
frequency of the excitation, then the recorded
intensity frame by a CCD camera represents an
average (for a time period, e.g. 20 ms in the case that
the CCD frequency is 50 Hz) of a specific temporal
phase of the vibration. This can be shown mathe-
matically if we multiply Equation (5) with a periodic
delta function as follows:
I
ST
¼
Z
T
CCD
IðtÞ
X
þ1
n¼1
dðt nTÞ
"#
dt (6)
where T
CCD
and T are the periods of the image col-
lection and vibration respectively. Considering that
the illumination (or mathematically the application
of the delta function) has a phase difference /
ST
with
the vibration, Equation (6) can be expanded as fol-
lows:
where I
min
and I
max
are constant intensity values for
each point, representing full destructive and con-
structive interference, respectively, between the two
superposed light sources. The key factor in Equation
(7) is A, which can be further developed as:
A ¼
Z
T
CCD
cos
4p
k
@d
0
@x
d
x
cosðxt þ/
ST
Þþð/
0
/
0
0
Þ
X
þ1
n¼1
dðt nTÞ
"#
dt
¼
X
þ1
n¼1
cos
4p
k
@d
0
@x
d
x
cosðxnT þ /
ST
Þþð/
0
/
0
0
Þ
()
¼
X
þ1
n¼1
cos
4p
k
@d
0
@x
d
x
cos/
ST
þð/
0
/
0
0
Þ
()
(8)
which proves that the collected light pattern does not
vary with time and corresponds to a frozen image of
the vibration at phase /
ST
. The signal acquisition for
a single pixel of the CCD recording a purely sinu-
soidal vibration is depicted in Figure 2.
Figure 2: Sampling function (stroboscopic illumination triggered at the fundamental frequency 200 Hz), normalised intensity
variation (Equation (18)) and recorded signal over one period (20 ms for a CCD with frequency of 50 Hz) for a single pixel taking
into account 3% of noise
I
ST
¼
Z
T
CCD
(
u
2
0
þ u
02
0
þ u
0
u
0
0
cos
"
4p
k
@d
0
@x
d
x
cosðxt þ /
ST
Þþð/
0
/
0
0
Þ
#)
X
þ1
n¼1
dðt nTÞ
"#
dt
¼
I
max
þ I
min
2
þ
I
max
I
min
2
Z
T
CCD
cos
4p
k
@d
0
@x
d
x
cosðxt þ /
ST
Þþð/
0
/
0
0
Þ
X
þ1
n¼1
dðt nTÞ
"#
dt
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
A
(7)
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd j Strain (2008) 44, 398–408 401
G. Kalogiannakis et al. : SHECROSSMOD
Stroboscopic dynamic shearography: nonlinear
case
In the case of nonlinear vibration dynamics, in spite
of purely sinusoidal excitation, an object vibrates at
multiple frequencies as a result of clapping and/or
cross-modulation. For simplicity let us just consider
the vibration at two different frequencies x
1
and x
2
.
Equation (4) becomes:
U ¼ u
0
exp i/
0
þ i
4p
k
½d
1
cos x
1
t þ d
2
cos x
2
t
(9)
Let us now consider that we trigger the stroboscope
at frequency x
1
. In this case, we practically apply a
periodic delta function synchronised to the first fre-
quency:
I
ST
¼
Z
T
CCD
IðtÞ
X
þ1
n¼1
dðt nT
1
Þ
"#
dt (10)
As a result, Equation (7) is transformed to:
I
ST
¼
Z
T
CCD
(
u
2
0
þu
02
0
þu
0
u
0
0
cos
"
4p
k
d
x
@d
1
@x
cosðx
1
t þ/
1
ST
Þþ
@d
2
@x
cosðx
2
t þ/
2
ST
Þ
!
þð/
0
/
0
0
Þ
#)
X
þ1
n¼1
dðt nT
1
Þ
"#
dt (11)
Similarly, the key factor is:
A¼
Z
T
CCD
X
þ1
n¼1
dðt nT
1
Þcos
"
4p
k
d
x
@d
1
@x
cosðx
1
t þ/
1
ST
Þ
þ
@d
2
@x
cosðx
2
t þ/
2
ST
Þ
!
þð/
0
/
0
0
Þ
#
dt
¼
X
n
CCD
n¼n
0
cos
"
4p
k
d
x
@d
1
@x
cosðx
1
nT
1
þ/
1
ST
Þ
þ
@d
2
@x
cosðx
2
nT
1
þ/
2
ST
Þ
!
þð/
0
/
0
0
Þ
#
¼
X
n
CCD
n¼n
0
cos
4p
k
d
x
@d
1
@x
cos/
1
ST
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
A
1
þ
@d
2
@x
cos 2pn
x
2
x
1
þ/
2
ST
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
A
2
0
B
B
B
@
1
C
C
C
A
2
6
6
6
4
þð/
0
/
0
0
Þ
#
(12)
which could be separated in the factor A
1
associated
with the frequency x
1
that is constant and the factor
A
2
associated with the frequency x
2
which oscillates
with a frequency of x
2
/x
1
. This latter factor can also
be constant if
x
2
¼ k x
1
; k 2 N (13)
In Figure 3, one can see a simulation example. A
cross-modulation signal is generated from the mixing
of two waves at 200 and 800 Hz. Assuming a CCD
frequency of 50 Hz and a noise level (in practical
situations noise can originate from optical and elec-
tronic sources) of 3%, the normalised intensity vari-
ation is drawn as a function of time. In Figure 3A, the
(A) (B)
Figure 3: Normalised intensity variation for the cross-modulation signal (200 and 800 Hz) and the recorded signal over one
period (20 ms for a CCD with frequency of 50 Hz) for different sampling frequencies: (A) triggered at the ‘pump’ frequency 200 Hz
and (B) triggered at the mixed frequency 600 (800 ) 200) Hz
402 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd j Strain (2008) 44, 398–408
SHECROSSMOD
: G. Kalogiannakis et al.