PREDICTION OF ULTRASONIC FIELDS INTO COMPOSITE
MULTI-LAYERED STRUCTURES: HOMOGENIZATION
APPROACH FOR THE DIRECT FIELD AND STATISTICAL
APPROACH FOR THE INNER REFLECTIONS
N. Gengembre1, P. Calmon, O. Petillon2 and S. Chatillon1
'CEA / LIST, CEA-Saclay, bat. 611, 91191 Gif-sur-Yvette cedex, France.
EADS-CCR BP76, 12 rue Pasteur, 92152 Suresnes cedex, France.
2
ABSTRACT. The aim of this work is to model the behavior of ultrasonic waves into multi-layered
components. The propagation of the direct field transmitted through the inner interfaces is predicted
by applying an homogenization approach, in which the parallel layers are replaced by a homogeneous
material with the proper stiffness constants. The noise due to inner reflections or interaction with
porosities is modeled in a second part by means of a statistical approach.
INTRODUCTION
The simulation of ultrasonic non destructive testing is more and more used in the
aircraft industry in the aim of improving the analysis of results, of optimizing the testing
configuration or of considering the controllability of a component in the early stage of its
design. This paper describes a study dedicated to the modeling of the ultrasonic
propagation in carbon-epoxy composite, a material of great importance in the aircraft
context. This work has been made jointly by CEA and EADS CCR, partly in the
framework of a Brite program.
The thickness and the complexity of carbon-epoxy components in aircraft industry
are continuously increasing. The parts are no longer quasi-isotropic and exhibit nonparallel faces. The geometry makes it necessary to rely on modeling tools to explain the
ultrasonic wave behavior. Therefore, a realistic simulation of the inspection of carbonepoxy components requires to take into account simultaneously the effects of the (parallel
or non-parallel) layers constituting the material and the CAD geometry of the component.
To succeed in such a challenge, the application of semi-analytical models is required since
intensive computations are required to optimize a design or an inspection. From several
years, CEA has developed such NOT modeling tools which are gathered in the software
platform CIVA [1]. Amongst them, the Champ-Sons model, based upon the so-called
pencil method is dedicated to the computation of ultrasonic beams transmitted by
transducers (either monolithic or phased-array [2]) in anisotropic or/and heterogeneous
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
957
materials [3] and CAD geometries [2]. This work shows how the Champ-Sons model has
been adapted in order to deal with multi-layered structures.
In the first part of the paper, we present a homogenization method that allows to
evaluate the effective stiffness constants of a material constituted of parallel layers. We
then explain how this method can be used when the component contains areas with nonparallel plies. The homogenized material being inputted in Champ-Sons, beams
transmitted in real components can be computed. We present here some examples of such
computations.
Another phenomenon of interest is the backscattering noise arising from the
multilayered structure. The analysis of this noise can contribute to the evaluation of the
quality of the components since it is due partly to reflections on the layers interfaces and,
when existing, partly to the scattering on heterogeneity such as porosities. In the second
part of the paper we present a complementary model developed in the aim of predicting
these phenomena. The formulation is established as a function of a few number of
parameters such as the attenuation into the material, thickness of the layers and some
statistical parameters describing inner reflections and scattering.
HOMOGENIZATION OF THE STRUCTURE
Method
In this part, one aims at approximating a succession of parallel anisotropic layers
with one anisotropic medium, represented by its stiffness constants, called the effective
stiffness constants of the whole layers.
Different homogenization methods can be found in literature [4-5]. The method
developed in the configuration to be considered here is based on Postma's work [5], since
this author gives a particular attention to layered structures. This approach is here extended
to the case of anisotropic layers.
Following Postma, the procedure is performed considering the periodicity of the
component. Indeed, it is only performed on one pattern (period) of the component. For
instance, in figure 1 , a pattern constituted of three layers is extracted. In what follows, the
homogenization procedure is applied to such a pattern, then considering three layers, but it
is equivalent when more layers are to be considered. In figure 1, the axes are also defined:
the interfaces lay along xy-plane.
The effective stiffness constants are denoted as C*jkl. Hooke's law writes for the
overall volume:
where rfj and ekl are the global stresses and strains for the overall pattern. In what
follows, one aims at expressing these amounts as a function of the local stresses and strains
(for each layer).
Due to the conditions of continuity (the layers are supposed perfectly bonded), the
longitudinal strains along the directions parallel to the interfaces (i.e. x and y) are equal,
whereas along z, it is expressed as a function of the strains into each layer (denoted as e(^n ,
with m, n = x, y, z and / = 1, 2, 3) and of their thickness J/ (one will also introduce in the
following equations the thickness of the pattern d, so that ZflT = d ). One has:
958
y
y
x
zZ
(1)
'
(2)
x
(3)
Z
z
FIGURE 1.
1. Multilayered
pattern
(right).
FIGURE
Multilayeredspecimen
specimen(left)
(left)and
andone
onerepeated
repeated
pattern
(right).
(2) (2)
ε xx(1) = ε xx( 2 ) = ε xx( 3) = ε xx
o
ε yy(1) = εyy yy( 2) = yyε yy(3) = yyε yy
(3) (3)
d ε zz = d 1ε zz(1) + d 2 ε zz( 2 ) + d 3ε zz(3)
(4) (4)
=
For a perfectly bonded interface, the normal stress is equal for both media, and also
Forglobal
a perfectly
normal
equalas:for both media, and also
equals the
stress.bonded
It is theninterface,
possible the
to write
the stress
globalisstress
equals the global stress. It is then possible to write the global stress as:
(5)
τ zz(1) = τ zz( 2) = τ zz(3) = τ zz
dτ xx = d 1τ xx(1) + d 2τ xx( 2 ) + d 3τ xx( 3)
(6)
(5)
(6)
…
Let us also notice that Hooke's law holds for each layer separately, that is:
Let us also notice that Hooke’s law holds for each layer separately, that is:
(«i) _ /-.(in) _(m)
~
(7)
(m) (m)
τ ij( m ) = C ijkl
ε kl for m = 1, 2, 3,
(7)
where C\^ is the stiffness constants for the layer m.
(m)
where CInijkl
the stiffness constants
for the
m.
theis homogenization
procedure,
onelayer
considers
in a first step that compression
stress isInapplied
to the whole pattern
representing
the layeredinmedium.
relations
(1-7)
the homogenization
procedure,
one considers
a first From
step that
compression
and after
some algebraic
manipulations,
a system of
independent
with the
From relations
(1-7)
stress
is applied
to the whole
pattern representing
thethree
layered
medium.equations
23
23
threeafter
variables
' is obtained.
Solving this
systemofallows
to express £*:equations
' as a function
and
some £^:
algebraic
manipulations,
a system
three independent
with the
2,3
of thevariables
local stiffness
the thickness
of each
layer,
and the
.
three
ε 1zz, 2,3constants,
is obtained.
Solving this
system
allows
to strains
expresse^ε 1zz,and
ase^
a function
In
a
second
step,
these
expressions
are
introduced
into
Hooke's
law,
what
leads,
.
of the local stiffness constants, the thickness of each layer, and the strains ε xx and ε yyby
identification,
to thestep,
values
of expressions
those amongst
effectiveinto
stiffness
constants
which
are by
In a second
these
are the
introduced
Hooke’s
law, what
leads,
involved whentoa the
compression
applied.theLet
us notice
here that
one can
take are
identification,
values of stress
those is
amongst
effective
stiffness
constants
which
advantage
on
the
symmetry
of
the
pattern
(including
the
symmetry
of
the
anisotropy
intotake
involved when a compression stress is applied. Let us notice here that one can
each
layer)
to
reduce
the
number
of
non-zero
stiffness
constants
to
be
found.
For
instance,
advantage on the symmetry of the pattern (including the symmetry of the anisotropy into
a pattern constituted of two layers with equal thickness and with the same medium with
each layer) to reduce the number of non-zero stiffness constants to be found. For instance,
a pattern constituted of two layers with equal thickness and with the same medium with
(a)
(b)
FIGURE 2. Two components with
(b) with non parallel layers.
(a) complex geometry, (a) with parallel layers,(b)
Dotted lines: orientation of the layers.
FIGURE 2. Two components with complex geometry. (a) with parallel layers. (b) with non parallel layers.
Dotted lines: orientation of the layers.
959
III
II
II Arrows: z-axisHI
Same component as I2b after homogenization.
of anisotropy system.
I
FIGURE 3.
FIGURE 3. Same component as 2b after homogenization. Arrows: z-axis of anisotropy system.
tetragonal symmetry with orientations 0° and 90° has a global tetragonal symmetry, so that
only
7 of thesymmetry
21 constants
to be evaluated.
In has
order
to recover
the remaining
effective
tetragonal
withhave
orientations
0° and 90°
a global
tetragonal
symmetry,
so that
stiffness
constants,
shear
stresses
are
then
applied
and
the
same
procedure
is
performed.
only 7 of the 21 constants have to be evaluated. In order to recover the remaining effective
Once
a succession
of parallel
layers
has been
described
by meansis of
its effective
stiffness
constants,
shear stresses
are then
applied
and the
same procedure
performed.
stiffness constants,
it can be of
replaced
a homogeneous
medium.by
If means
a component
contains
Once a succession
parallelbylayers
has been described
of its effective
some
areasconstants,
with non it
parallel
such
described in figure
2b, Ifthea component
component contains
is firstly
stiffness
can belayers,
replaced
by as
a homogeneous
medium.
divided
in several
in layers,
which such
pliesas can
be considered
parallel.
The
some areas
with nonareas
parallel
described
in figure 2b,asthenearly
component
is firstly
homogenization
method
is then
appliedplies
successively
on each area.
The resulting
divided in several
areas
in which
can be considered
as nearly
parallel.virtual
The
component
is constituted
smallapplied
numbersuccessively
of homogeneous
anisotropic
homogenization
methodofisathen
on each
area. Thevolumes.
resulting virtual
component is constituted of a small number of homogeneous anisotropic volumes.
Example of Application
Example of Application
This approach has been applied in the aim of predicting the controllability of non
This(60
approach
has been
applied
the aim of predicting
the controllability
of non
planar thick
to 90 plies)
carbon
epoxyincomponent.
The geometry
of these components
planaron
thick
(60 2topresent
90 plies)
carbon
epoxy misalignment
component. The
geometry
of thesebycomponents
shown
figure
a four
degrees
which
is achieved
modifying
shown on figure
2 present
four Due
degrees
misalignment
which
is achievedregion
by modifying
continuously
the number
of aplies.
to this
process, the
misalignment
contains
continuouslydisorientated
the number layers.
of plies. Due to this process, the misalignment region contains
continuously
continuously
From a disorientated
criteria basedlayers.
on the disorientation range, the misalignment region (Region
From
the different
disorientation
range,
misalignment
II in figure
3) aiscriteria
dividedbased
into on
three
volumes
onthe
which
has been region
carried(Region
out the
II in figure 3) algorithm.
is divided into
different
volumes materials
on which differ
has been
out the
homogenization
The three
resulting
homogenized
fromcarried
each other
by
algorithm. axis.
The resulting homogenized materials differ from each other by
thehomogenization
orientation of anisotropy
the orientation
of anisotropy
axis.
Computations
of the ultrasonic
beam transmitted by a L0 5MHz focused probe
Computations
of
the
ultrasonicinbeam
transmitted
by a LO
5MHz
probe
have been performed by Champ-Sons
various
configurations.
One
resultfocused
is shown
on
have been performed by Champ-Sons in various configurations. One result is shown on
(a)
(b)
(a)
(b)
FIGURE
4.
(a)
Computation
of
the
field
radiated
into
component
2a.
(b)
idem
component 2b.
2b.
FIGURE 4. (a) Computation of the field radiated into component 2a. (b) idem for
for component
960
Figure 4b. Unsurprisingly, the surface misalignment induces a deviation of the beam. In
order
to evaluate
the relativethe
influence
this effect and
of the
internal disorientation
Figure
4b. Unsurprisingly,
surface of
misalignment
induces
a deviation
of the beam. Inof
theorder
plies,to beam
computations
have
also
been
performed
on
the
simplified
component
evaluate the relative influence of this effect and of the internal disorientation
of
schematized
on
Figure
2a
in
which
the
plies
are
parallel.
The
simulated
beam
is
shown on
the plies, beam computations have also been performed on the simplified component
Figure
4a. Theoncomparison
two
a curvature
the beam
schematized
Figure 2a inbetween
which thethe
plies
areresults
parallel.exhibits
The simulated
beam of
is shown
on
reflecting
the
internal
plies
disorientation.
This
result
is
of
practical
importance
Figure 4a. The comparison between the two results exhibits a curvature of thesince
beamit
demonstrates
the plies
beamdisorientation.
steering dueThis
to result
the surface
misalignment
is since
partially
reflecting thethat
internal
is of practical
importance
it
compensated
by that
the plies
compensation
increases with
the depth
demonstrates
the disorientation
beam steeringeffect.
due This
to the
surface misalignment
is partially
andcompensated
it can be seen
onplies
this disorientation
example thateffect.
the wave
normally
the backwall
of the
by the
This reaches
compensation
increases
with the depth
component
at on
a depth
ranging between
30 andreaches
40 mm.normally the backwall of the
and it canlocated
be seen
this example
that the wave
component located at a depth ranging between 30 and 40 mm.
NOISE PREDICTION
NOISE PREDICTION
The previously described model aims at predicting the transmitted beam. Such a
described
modelsignals
aims atarising
predicting
Such a
result canThe
be previously
used in order
to predict
fromthethetransmitted
structure.beam.
Nevertheless,
result
can
be
used
in
order
to
predict
signals
arising
from
the
structure.
Nevertheless,
predictions of real signals received by the probe require a complementary model in order to
predictions
of real signalsofreceived
by thesources
probe require
a complementary
model
in order toon
simulate
the contributions
the different
of noise
which are internal
reflections
simulate
the
contributions
of
the
different
sources
of
noise
which
are
internal
reflections
on
the layers, distributed scatterers (such as porosities) and electronic gain.
the layers, distributed scatterers (such as porosities) and electronic gain.
Analysis
Analysis
The first step of our present study has been to analyze some experimental A-scans
The first step of our present study has been to analyze some experimental A-scans
in order to have a good description of the noise due to these inner reflections. In figure 5, a
in order to have a good description of the noise due to these inner reflections. In figure 5, a
signal received by a 5Mhz transducer is shown. The inspected component is a carbon
signal received by a 5Mhz transducer is shown. The inspected component is a carbon
epoxy plate immersed in water. One can distinguish two main echoes, the first one being
epoxy plate immersed in water. One can distinguish two main echoes, the first one being
duedue
to tothethereflection
the second
second one
one being
beingthe
the
reflectionatatthe
thecoupling
coupling / / sample
sample interface,
interface, and
and the
backwall
echo.
Between
these
two
main
echoes,
many
echoes
of
lower
amplitude
are
backwall echo. Between these two main echoes, many echoes of lower amplitude are
detected.
The
wavelet
analysis
of
this
signal
has
been
performed
in
order
to
analyze
the
detected. The wavelet analysis of this signal has been performed in order to analyze the
frequency
range
of
these
different
echoes
and
is
shown
in
figure
5b.
One
can
observe
that
frequency range of these different echoes and is shown in figure 5b. One can observe that
thethe
backwall
echo
sample echo,
echo,due
duetotothe
the
backwall
echohas
hasa alower
lowercenter
centerfrequency
frequency than
than the
the coupling
coupling // sample
attenuation
into
the
sample,
higher
for
high
frequencies.
The
frequency
of
the
contributions
attenuation into the sample, higher for high frequencies. The frequency of the contributions
of of
low
is, for
for the
the same
samereason,
reason,
lowamplitude
amplitudeappearing
appearingbetween
between these
these two
two echoes
echoes (noise)
(noise) is,
slightly
decreasing
as
time
grows
up.
But
one
can
also
observe
on
figure
5
that
a
part
the
slightly decreasing as time grows up. But one can also observe on figure 5 that a part ofofthe
noise
signal,
appearing
just
after
the
coupling
/
sample
echo
contains
higher
frequencies,
noise signal, appearing just after the coupling / sample echo contains higher frequencies,
2.d I/ c,
c, d:
d: thickness,
thickness,
centered
at at
5.65.6
MHz,
fr of
centered
MHz,that
thatisisthe
theresonant
resonantfrequency
frequency^
of one
one ply
ply (f
(fr r -= 2.d
(a)(a)
(b)(b)
FIGURE
ExperimentalA-scan
A-scanreceived
receivedfrom
from aa 5MHz
5MHz transducer
transducer (top)
(top) and
FIGURE
5. 5.Experimental
and its
its time
time / / frequency
frequency
representation
obtained
meansofofwavelet
wavelettransform
transform(bottom).
(bottom).
representation
obtained
byby
means
961
FIGURE 6. Experimental A-scan received from a 10MHz transducer
c: wavespeed) and that these contributions disappear after a while.
On figure 6, another experimental signal has been plotted in order to emphasize
that, when the gain used in the DAC function (electronic gain as a function of time) is
high, electronic noise appears (at the end of the signal in figure 6).
Model
One assumes that the noise has three different origins: the structure (layers) giving
rise to structural noise, inner scatterers, such as porosities, giving rise to backscattered
noise, and the effects of the acquisition line giving rise to electronic noise. These three
different noises are modeled using a statistical approach.
The structural noise is modeled using the value of the attenuation coefficient a in
the component. This coefficient expresses the ratio between the amplitude of a
monochromatic plane wave (f. frequency) after a propagation along a distance z and its
amplitude without attenuation. One assumes:
(8)
4(0)
An inverse Fourier transform of this function gives the impulse response of a wave
reflected by an interface located at depth z and propagated back to the transducer. One also
considers that the part of the wave transmitted is represented by a reflection coefficient /?/ ,
that is a statistical coefficient to be calibrated. It represents the quality of the bond at the
interfaces.
If this procedure is applied at every interfaces, one obtains the overall impulse
response for the whole component, as plotted in figure 7 on which are also seen the surface
and backwall echoes. This figure also shows the result of the convolution of this impulse
response with the input signal and the application of the DAC.
The backscattering noise is modeled by considering porosities as reflectors with
varying reflectivity. The impulse response of such scatterers is simulated by a set of delta
functions at random times. The amplitude of each delta function is extracted from a
statistical gaussian distribution describing porosities reflectivity [6]. As for Ri9 the
distribution mean square has to be evaluated in a previous calibration phase. The impulse
response obtained by this method is also to be convoluted with the input signal and then
added to the structural noise.
To deal with the electronic noise, a white gaussian noise is then added to the
resulting signal.
962
(a)
(a)
(b)
(b) ill
FIGURE
(b) After
After aa
FIGURE7.7.(a)(a)Impulse
Impulseresponse
responseofofa amultilayered
multilayered component
component with
with account
account of
of interfaces.
interfaces, (b)
convolution
convolutionwith
withthe
theinput
inputsignal
signaland
andapplication
application of
of the
the DAC
DAC function.
function.
Comparisons
been performed.
performed.
Comparisonsbetween
between simulated
simulated and
and experimental
experimental signals
signals have
have been
The
show that
that this
this
Theexample
exampleofofa a10MHz
10MHztransducer
transducerisisgiven
given in
in figure
figure 8.
8. These
These comparisons
comparisons show
approach
of the
the noise
noise
approachisisable
abletotopredict
predictthe
the main
main features
features of
of the
the different
different contributions
contributions of
received
In particular
particular we
we
receivedduring
duringthe
theinspection
inspectionofofaacarbon
carbon epoxy
epoxy multiplayer
multiplayer component.
component. In
can
resulting from
from the
the
cansee
seeononFigure
Figure8 8that
thatthe
thedecreasing
decreasingamplitude
amplitude of
of the
the internal
internal echoes,
echoes, resulting
superimposition
superimpositionofofseveral
severalcomplex
complexphenomena,
phenomena, isis quite
quite well
well predicted.
predicted.
(a)
^y-A^^w^
(b)
FIGURE8.8.(a)(a)Simulated
Simulatedand
and(b)
(b)experimental
experimentalsignals
signalsreceived
received by
by aa 10
10 MHz
MHz probe.
probe.
FIGURE
963
CONCLUSION
hi this paper we have presented a modeling study dedicated to the simulation of
ultrasonic inspection of carbon epoxy multilayered components. An approach in order to
predict the ultrasonic beams transmitted into such components has been firstly reported.
This approach is based on a homogenization method which aims at replacing a succession
of layers with only one homogeneous and anisotropic medium. After this procedure, the
ultrasonic field radiated into multi-layered components is computed by the Champ-Sons
model. This approach allowed us to study ultrasonic propagation in components involving
non-parallel internal plies and complex geometries. A complementary model dedicated to
the prediction of signals received by the probe when inspecting multilayered components
has been also reported. It includes the simulation of structural noise, due to inner
reflections, backscattered noise, due to scatterers such as porosities and electronic noise.
Comparisons between simulated and experimental results show a good agreement.
REFERENCES
1. Calmon P., Lhemery A., Lecoeur-Taibi I. and Raillon R., "Integrated models of
ultrasonic examination for NDT expertise", in Review of progress in QNDE, Vol.16,
eds. D. O. Thompson and D. E. Chimenti, Plenum, New-York, 1997, pp. 1861-1868.
2. Mahaut S., Roy O., Chatillon C. and Calmon P., "Modelling and application of phased
array techniques dedicated to complex geometry inspection", in Review of progress in
QNDE, Vol. 21, op.cit., 2001, pp. 894-901
3. Gengembre N. and Lhemery A., "Calculation of wideband ultrasonic fields radiated by
water-coupled transducers into heterogeneous and anisotropic media", in Review of
progress in QNDE, Vol. 19, op.cit., 1999, pp. 977-984.
4. Achenbach J.D., A theory of elasticity with microstructure for directional'ly reinforced
composites, Springer Verlag, New-York, 1975.
5. Postma G. W., Geophysics 20, 780-806, 1955.
6. Chatillon S., Poidevin C., Gengembre N. and Lhemery A., "Simplified modeling of
backscattered noise and attenuation phenomena for quantitative performance
demonstration of UT methods", in Review of progress in QNDE, Vol. 22, op. cit., eds.
D. O. Thompson and D. E. Chimenti, these proceedings.
964