Identification of the Elastic Properties of Isotropic
and Orthotropic Thin-Plate Materials with the Pulsed
Ultrasonic Polar Scan
M. Kersemans & A. Martens & N. Lammens &
K. Van Den Abeele & J. Degrieck & F. Zastavnik &
L. Pyl & H. Sol & W. Van Paepege m
Received: 20 November 2013 /Accepted: 4 February 2014 /Published online: 1 March 2014
#
Society for Experimental Mechanics 2014
Abstract Already in the early 1980’s, it has been conjectured
that the pulsed ultrasonic polar scan (P-UPS) provides a
unique fingerprint of the underlying mechanical elasticity
tensor at the insonified material spot. Until now, that premise
has not been thoroughly investigated, nor validated, despite
the opportunities this would create for NDT and materials
science in general. In this paper, we report on the first-ever
implementation of an inverse modeling technique on the basis
of a genetic optimization scheme in order to extract quantita-
tive information from a P-UPS. We validate the optimization
approach for synthetic data, and apply it to experimentally
obtained polar scans for annealed aluminum, cold rolled
DC-06 steel as well as for carbon fiber reinforced plastics.
The investigated samples are plate-like and do not require
specific preparation. The inverted material characterist ics
show good agreement with literature, micro-mechanical
models as well as with results obtained through conventional
testing procedures.
Keywords Ultrasound
.
Polar scan
.
Metals and composites
.
Elastic properties
.
Characterization
Introduction
The high strength-to-weight ratio, in combination with a tun-
able stiffness, make composite materials very attractive for a
range of high-tech applications, which include wind turbine
blades, primary components in aerospace industry, etc. For
monitoring and inspecting these critical structures, a variety of
approaches are available [1–5], among which the most wide-
spread concerns the application of ultrasound. Besides the
classical ultrasonic C-scan, which has already proven its use-
fulness in detecting defects, delaminations as well as material
heterogeneities, several more advanced techniques can quan-
titatively monitor the mechanical performance in order to
ensure the designed functionality of a composite structure.
At present, (nonlinear) resonant ultrasonic spectroscopy
((N)RUS) is a well-established technique to infer elasticity
constants [6], and even monitor damage progression [7–9], of
anisotropic materials. However, (N)RUS is a resonance based
technique, in which the specifically prepared sample needs to
be mounted in a device, making it useless for in situ determi-
nation of the material characteristics. The latter restriction also
holds for techniques which are based on time measurements
of bulk waves because measurements have to be taken at
specific material orientations [10–15]. Besides, for anisotropic
media the propagation- and energy velocity are decoupled,
making the correct recording and subsequent interpretation of
experimental data a rather difficult task. Plane wave transmis-
sion characteristics have been used by various research teams
to extract the (visco)elastic material properties [16, 17].
Basically, the sample is insonified at a few well-chosen
oblique incidence angles with a broadband ultrasonic pulse
which resembles a plane wave [18]. The transmitted wave
field is recorded in time, normalized to the reference wave
field, and further analyzed in the frequency domain. In this
way the transmission coefficient is obtained as a function of
M. Kersemans (*)
:
N. Lammens
:
J. Degrieck
:
W. Van Paepege m
Department of Materials Science and Engineering, Ghent University,
Technologiepark-Zwijnaarde 903, 9052 Zwijnaarde, Belgium
e-mail: mathias.kersemans@ugent.be
A. Martens
:
K. Van Den Abeele
Department of Physics, Catholic University of Leuven, Campus
Kortrijk - KULAK, Etienne Sabbelaan 52, 8500 Kortrijk, Belgium
F. Zastavnik
:
L. Pyl
:
H. Sol
Department Mechanics of Materials and Construction, Vrije
Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Experimental Mechanics (2014) 54:1121–1132
DOI 10.1007/s11340-014-9861-7
the frequency. The resulting transmission coefficient has sharp
minima which relate to the condition for efficient stimulation
of Lamb waves, while the global level of the transmission
coefficient is a measure for the attenuation. Another closely
related technique which gained a lot of popularity over the last
years concerns the application of guided waves, such as Lamb
waves, Rayleigh waves, etcetera, which probe the structure
under investigation. The stimulated guided wave interrogates
the structure upon propagation, hence the recorded signal
contains a watermark of the material characteristics [19–21].
The main disadvantage of such a guided wave based approach
concerns the complexity of the data acquisition and the exten-
sive post-processing in order to obtain the envisioned results,
especially when the investigated structure is for example not-
flat/curved [22–24]. Besides, the guided wave technique pro-
vides a global representation of the material characteristics,
while it is often needed to obtain the material characteristics at
a local material spot.
Already in the early 80’s, the ultrasonic polar scan (UPS)
technique was introduced [25] as a promising tool for NDT
and material research. UPS’s are created by insonifying a
material spot for a 2D range of oblique incidence angles,
and simply gathering the tra nsmitted (or if necessary the
reflected) maximum pulse amplitude [25–27]. A schematic
of the method is shown in Fig. 1(a). By mapping the recorded
amplitudes in polar representation (φ,θ), a polar scan image is
formed in which intriguing contours emerge (see Fig. 1(b)).
Each point in a UPS image corresponds to a unique incidence
angle ψ(ϕ,θ), while the assigned color pigment is a measure
for the transmitted maximum amplitude. The characteristic
contours in a UPS image are linked to the generation of
critically refracted (quasi-)bulk waves in the case of sound
pulses [28], and thus are from a physics point of view directly
connected to the mechanical properties of the sample material
at the targeted spot. In the remainder of this study, we denote a
pulsed ultrasonic polar scan by P-UPS. More recently, it has
been shown that the operational regime of the P-UPS can be
expanded to the recording of the associated time-of-flight
(TOF) value [29, 30], to the recording of backscattered waves
[31] as well as to the analysis of the associated phase in case of
a harmonic version of the polar scan (H-UPS) [27], in order to
gain supplementary information of the insonification spot. In
this paper however, we limit ourselves to amplitude recorded
P-UPS results in a transmission setup. Since a single material
spot is interrogated in an omnidirectional way, no prior knowl-
edge about the material is required to obtain the envisioned
data. The very simple data-acquisition in which only the
maximum transmission pulse amplitude needs to be stored
as well as the straightforward post-processing, makes it a very
appealing and simple technique to extract material character-
istics. Especially when considering the enormous amount of
redundant data in a P-UPS experiment, from which an inver-
sion procedure would obviously benefit to efficiently extract
material properties with a high degree of accuracy. Finally, it is
worth noting that numerical computations indicate that the
P- UPS technique works equally well in a reflection setup
[28].
It has been already conjectured in several studies that the
characteristic contours in a P-UPS image are a unique encryp-
tion of the local elastic material properties [25, 26, 28, 29,
32–35]. Nevertheless this statement has not yet been compre-
hensively investigated, nor verified, which immediately de-
fines the scope of the present paper. We investigate numeri-
cally as well as experimentally the possibility to extract the
local material characteristics from an amplitude recorded P-
UPS.
The next section gives a short description of the imple-
mented system identification procedure, which is based on an
optimization procedure by means of a genetic algorithm (GA).
Inversion results are first presented and discussed for synthetic
data, in order to explore the opportunities as well as the
limitations of the method for material characterization.
Fig. 1 Schematic of the UPS principle (a), numerically computed P-UPS for a [0]
8
carbon/epoxy laminate (b)
1122 Exp Mech (2014) 54:1121–1132
Finally, the system identification procedure is applied to ex-
perimentally obtained P-UPS exp eriments for sev eral
annealed aluminum samples in different mechanical
health, cold-rolled DC-06 steel samples and an autoclave
manufactured [0]
8
carbon fiber reinforced plastic plate.
Optimization Procedure
Since the characteristic contours in a P-UPS image are a
representation of the in-plane critical bulk wave angles for all
polarization states [28], it is clear that the inversion procedure
should be based on the well-known Christoffel equation.
Indeed, the characteristics of plane wave propagation in an
infinite elastic anisotropic solid are effectively described by the
Christoffel equation (Einstein summation convention) [36].
ρω
2
δ
im
− C
iklm
k
k
k
l
u
m
¼ 0 ð1Þ
with ρ the density, ω the circular frequency, δ the Kronecker
delta, C the elasticity tensor, k the unit wave vector and u the
polarization of the mechanical wave. This equation leads to an
orthogonal classical eigenvalue problem for which three inde-
pendent solutions exist in case of a symmetry higher than or
equal to monoclinic symmetry. The eigenvalue problem is
solved by demanding nontrivial solutions, which is equivalent
to setting the determinant of the coefficient matrix equal to zero:
ρω
2
δ
im
− C
iklm
k
k
k
l
¼ 0 ð2Þ
With this, the computed eigenvalues can be entered back
into the system of original homogeneous equations to calcu-
late the eigenvectors and confirm the wave type and character,
i.e. pure or quasi. Application of Snell-Descartes’ law for
externally borne sound finally leads to the determination of
the critical bulk wave angles which are then compared with
the characteristic contours in a P-UPS image. By selecting an
appropriate set of elastic constants, the computed in-plane
bulk wave angle profiles can then be matched with the P-
UPS contours.
In general, composite materials are anisotropic of nature, in
which orthotropy can be considered the most common sym-
metry class. A full description of the orthotropic nature can be
given by knowledge of nine elastic constants. Hence it is clear
that an appropriate set of these nine constants, to obtain a good
fit with the contours hidden in the P-UPS image, cannot be
simply obtained by trial and error. This is especially true
considering that the elastic constants describe a highly
nonlinear space. In this paper, an optimization scheme is
employed which relies on the principle of Darwin, i.e. an
abstraction is made in terms of a population in which the
hierarchy is determined by the survival of the fittest (see
Fig. 2). A set of different C
ij
’s corresponds to a complete
population in the genetic algorithm (GA), while a single
elasticity tensor C
ij
corresponds to an individual in the popu-
lation. The choice for a genetic optimization procedure is
mainly triggered by its great flexibility for optimizing nonlin-
ear multi-dimensional problems, its capability to converge to
the near optimum of the problem regardless the initial guess
and the fact that a GA can be easily parallelized, reducing the
computational effort. An even more important advantage of
GA concerns its capability to not remain trapped in a subop-
timal solution since a population covers different parts of the
parameter space. GA’s have already been successfully
employed in various studies for inverting (synthetic) wave
velocity data in order to obtain a measure of the mechanical
properties [12, 14, 20, 37].
Basically we start with a random initial population,
consisting of 100 individuals, i.e. 100 unique elasticity tensors
C
ij
, from which the corresponding critical bulk wave profiles
are computed through application of Christoffel’sequationin
combination with the Snell-Descartes law. The parameter
space needs bounds in which the GA has a complete freedom
of selectivity in order to obtain the optimized set of elasticity
constants C
ij
. Of course, these bounds should be chosen as
narrow as possible to speed up the inversion process, while
assuring that the space contains the optimal solution.
Typically, we set the bounds at ±90 % of the estimated
elasticity constants C
ij
to make sure that the parameter space
contains the optimal solution. The increment at which the
elasticity parameters can be updated, is chosen fairly small
in order to improve the accuracy. The computed critical bulk
wave angle profiles are then simply superposed on the P-UPS
from which a measure of their fitness is obtained by summing
the interpolated P-UPS amplitudes at the exact coordinates of
the computed curves, and this for all polarization states
X
k
X
l
A
I
ϕ
k
; θ
l
k;Christ
ð3Þ
with A
I
(ϕ
k
,θ
k
l
) the interpolated transmitted pulse amplitude at
coordinates (ϕ
k
,θ
k,Christ
l
) in which ϕ
k
, respectively θ
k,Christ
l
,
represents the polar direction, respectively the computed crit-
ical bulk wave angle for a given elasticity tensor C
ij
and l=
QL,QSH,QSV denotes the quasi-longitudinal, the quasi-shear
horizontal and the quasi-shear vertical polarization state. This
is illustrated in Fig. 3 for an arbitrary elasticity tensor C
ij
.
Since the computed profiles should merge to the low-
amplitude contours in the P-UPS image in order to obtain a
qualified elasticity tensor C
ij
, it is clear that the above criterion
has to be minimized:
min
X
k
X
l¼QL;QSH;QSV
A
I
ϕ
k
; θ
l
k;Christ
!
ð4Þ
Exp Mech (2014) 54:1121–1132 1123
The fact that the superimposed lines do not fully match the
P-UPS contours in Fig. 3 clearly reveals that th e current
elasticity tensor C
ij
cannot be regarded as the optimal elasticity
tensor C
ij
.
It is noted that the P-UPS contours are also characterized by
a high local curvature value, hence the fitness criterion can
also be adapted in order to maximize the summed curvature
values. In fact, this criterion is also applied to the P-UPS
data of the m etallic samples because of the accelerated
convergence.
The above procedure is executed for each elasticity tensor
C
ij
in the current population, the obtained fitness values for
that population are then sorted in descending order. After this,
the current population generates the next population by pro-
ducing descendants. This step is basically dictated by three
descendant-generation rules. First, those individuals having
superior fitness values are directly recruited in the next gen-
eration according to the principle of elitism, thus forming the
elite children (20 %). Second, crossover children are formed
by randomly mixing the characteristics of any two individuals
(60 %). Lastly, the next population is further complemented
with random mutations of an individual by randomly flipping
bit values from 0 to 1 and vice versa, thus resulting in a set of
mutated individuals (20 %). Once the nex t generation is
created, the fitting procedure is repeated in order to obtain a
fitness value of the updated generation. This loop is repeated
for many updated populations, until convergence is obtained.
The stopping procedure is activated when two criterions are
satisfied at the same time. The variation of the fitness value of
both the current best individual and the mean of the population
should be sufficiently small over ten subsequent generations.
Once these conditions are fulfilled, the procedure is stopped
and the optimized elasticity tensor C
ij
is obtained.
Nevertheless, our first inversion results revealed that our
basic assumption, being the fact that the P-UPS contours
provide a one-to-one relationship with the in-plane critical
bulk wave angles [28], is not exactly valid. This is numerically
demonstrated in Fig. 4 for a hypothetical transversal isotropic
material. The P-UPS simulation is performed according to the
recursive stiffness matrix method [27, 38, 39]. The spectral
frequency content of a typical ultrasonic broadband pulse is
accounted for by means of a Fourier integral. It can be seen
that the superposed critical bulk wave angle profiles slightly
deviate from the characteristic contours in the numerically
computed P-UPS, although both have been computed with
Fig. 2 Schematic of the optimi-
zation algorithm
Fig. 3 P-UPS experiment for a [0]
8
carbon/epoxy laminate. The critical
bulk wave angle profiles, computed for an arbitrary elasticity tensor C
ij
,
are superposed
1124 Exp Mech (2014) 54:1121–1132
the same set of material parameters (listed in Table 1).
Especially for the inner contour, which relates to the stimula-
tion of the in-plane propagating quasi-longitudinally (QL)
polarized bulk wave, and in extension to the Young’smodulus
E
ii
, this discrepancy can be clearly observed along φ=0° (see
Fig. 4(b)). The position of this P-UPS minimum is rather
dictated by that incident angle θ where the shear horizontal
wave component in the solid becomes dominant over the
longitudinally polarized component [40]. Although the angu-
lar deviation can be considered to be small, it is critical to
account for in order to obtain a correct and accurate inversion
of the elasticity tensor C
ij
from a P-UPS image. Especially
when considering that the here considered composite material
has a very high mechanical stiffne ss along φ=0°, which
implies that the inner contour has a small incident angle θ.
According to Snell-Descartes’ law, any small change in inci-
dent angle then results in a substantial change in wave speed
and in extension to the underlying mechanical stiffness.
Therefore, the system identification procedure on the basis
of the Christoffel solutions needs an extension to account for
this discrepancy.
The extension of the inversion procedure basically relies on
the same scheme as described in Fig. 2. Though, an initial
population is defined based on the elasticity tensor C
ij
obtain-
ed in the first inversion procedure on the basis of t he
Christoffel solutions. From this set of elasticity tensors C
ij
,
P-UPS sectors at a few fixed polar angles φ are computed. The
P-UPS simulations are performed according to the recursive
stiffness matrix method [27, 38, 39]. By allowing small per-
turbations in the elasticity tensor C
ij
, the local minima in the
computed P-UPS sections are matched with the observed
minima in the P-UPS experiment. The main reason why only
a few sectors are considered, instead of the full P-UPS image,
is mainly triggered by computation constraints. Indeed, the
Fig. 4 Simulated P-UPS at fd=
5 MHz×1 mm for a hypothetical
orthotropic material with the
computed critical bulk wave an-
gle profiles superposed for the
range ϕ∈[0 °,180°] (a)and
magnification to accentuate the
discrepancy between the position
of the P-UPS contours and the
critical bulk wave angle
profiles (b)
Table 1 Inversion results for synthetic data: mean value ± standard deviation (relative error %). The last row is put in gray as it has been obtained through
symmetry considerations
Input
[GPa]
Output [GPa]
Scatter 0% Scatter 5% Scatter 15% Scatter 30%
C
11
117.43
117.46 ± 0.50
(+0.03%)
118.05 ± 3.05
(+0.53%)
114.99 ± 5.91
(-2.12%)
130.55 ± 23.93
(+10.05%)
C
12
= C
13
4.99
5.03 ± 0.07
(+0.96%)
5.05 ± 0.38
(+1.20%)
4.96 ± 0.70
(-0.54%)
5.76 ± 2.39
(+13.39%)
C
22
= C
33
13.94
13.96 ± 0.08
(+0.15%)
14.01 ± 0.10
(+0.50%)
13.97 ± 0.30
(+0.19%)
14.76 ± 1.32
(+5.57%)
C
44
3.90
3.91 ± 0.07
(+0.27%)
3.89 ± 0.06
(-0.28%)
3.92 ± 0.11
(+0.48%)
3.98 ± 0.29
(+1.95%)
C
55
= C
66
7.19
7.16 ± 0.00
(-0.44%)
7.13 ± 0.04
(-0.87%)
7.06 ± 0.13
(-1.82%)
6.62 ± 0.93
(-8.56%)
C
23
= C
22
- 2C
44
6.14
6.14 ± 0.21
(0.00%)
6.23 ± 0.08
(-1.47%)
6.13 ± 0.47
(-0.16%)
6.8 ± 0.89
(-10.75%)
Exp Mech (2014) 54:1121–1132 1125
![Fig. 1 Schematic of the UPS principle (a), numerically computed P-UPS for a [0]8 carbon/epoxy laminate (b)](/figures/fig-1-schematic-of-the-ups-principle-a-numerically-computed-37j55t3z.webp)
![Fig. 8 P-UPS recorded at fd=5 MHz×1.1 mm for an autoclave manufactured [0°]8 carbon/epoxy laminate (a) and after rotation over φfiber (b)](/figures/fig-8-p-ups-recorded-at-fd-5-mhzx1-1-mm-for-an-autoclave-1qwrz7js.webp)
