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Towards real-time assessment of anisotropic plate
properties using elastic guided waves
Nicolas Bochud, Jérôme Laurent, Francois Bruno, Daniel Royer, Claire Prada
To cite this version:
Nicolas Bochud, Jérôme Laurent, Francois Bruno, Daniel Royer, Claire Prada. Towards real-time
assessment of anisotropic plate properties using elastic guided waves. Journal of the Acoustical Society
of America, Acoustical Society of America, 2018, 143. �hal-02335140�
Towards real-time assessment of anisotropic plate properties using elastic guided
waves
Nicolas Bochud, Jérôme Laurent, François Bruno, Daniel Royer, and Claire Prada
Citation: The Journal of the Acoustical Society of America 143, 1138 (2018); doi: 10.1121/1.5024353
View online: https://doi.org/10.1121/1.5024353
View Table of Contents: http://asa.scitation.org/toc/jas/143/2
Published by the Acoustical Society of America
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Towards real-time assessment of anisotropic plate properti es
using elastic guided waves
Nicolas Bochud, J
er
^
ome Laurent, Franc¸ois Bruno, Daniel Royer, and Claire Prada
a)
Institut Langevin, ESPCI Paris, CNRS (UMR 7587), PSL Research University, 75005 Paris, France
(Received 5 December 2017; revised 29 January 2018; accepted 29 January 2018; published online
22 February 2018)
A method to recover the elastic properties, thickness, or orientation of the principal symmetry axes
of anisotropic plates is presented. This method relies on the measurements of multimode guided
waves, which are launched and detected in arbitrary directions along the plate using a multi-
element linear transducer array driven by a programmable electronic device. A model-based inverse
problem solution is proposed to optimally recover the properties of interest. The main contribution
consists in defining an objective function built from the dispersion equation, which allows account-
ing for higher-order modes without the need to pair each experimental data point to a specific
guided mode. This avoids the numerical calculation of the dispersion curves and errors in the mode
identification. Compared to standard root-finding algorithms, the computational gain of the proce-
dure is estimated to be on the order of 200. The objective function is optimized using genetic algo-
rithms, which allow identifying from a single out-of-symmetry axis measurement the full set of
anisotropic elastic coefficients and either the plate thickness or the propagation direction. The effi-
ciency of the method is demonstrated using data measured on materials with different symmetry
classes. Excellent agreement is found between the reported estimates and reference values from the
literature.
V
C
2018 Acoustical Society of America. https://doi.org/10.1121/1.5024353
[ANN] Pages: 1138–1147
I. INTRODUCTION
Elastic waves are frequently applied for nondestructive
material characterization and have been used with success in
the past decades. In particular, several attempts to identify
the anisotropic stiffness tensor have been made, which is an
essential task for modeling and evaluating the mec hanical
behavior of polycrystalline metals and composite materials.
Initially, the anisotropic elastic properties of a number of
materials have been determined through bulk waves mea-
surements.
1,2
Nonetheless, these experiments require the
measurements to be performed along many propagation
directions. In addition, for wave speed data processing, the
correct identification of the two transverse waves is not triv-
ial and some additional information is usually required for
this purpose.
3
Alternatively, there are several advantages in
using elastic guided waves (EGWs) for the nondestructive
evaluation of anisotropic materials.
4
Their dispersive and
multimodal nature is especially useful when it is desirable to
use wavelengths larger than the plate thickness or when it is
necessary to measure in-plane elastic properties,
5
as the
components of the elastic tensor affect each mode differently
and with different sensitivities.
6
To achieve the identification of anisotropic material
properties, two issues have to be addressed. The first one is
related to the ultrasonic measurement configuration and sig-
nal acquisition method, i.e., the use of both narrowband and
broadband excitations has been reported.
7
Second, a robust
model-based inverse procedure is required to extract the
elastic properties from the measured data.
8
Narrowband
signals are used to excite several pure modes by sweeping
the frequency and varying the transducer angle of incidence,
thus allowing the recording of phase velocities as discrete
points over a broad frequency range by means of the phase
shift method.
5
This approach can be implemented in many
ways using, for example, variable angle wedges in pitch–
catch configuration,
5
air-coupled transducers,
9,10
oblique
insonification of a test specimen that is fully immersed in
water [so-called leaky Lamb wave (LLW) technique],
6,11,12
or a piezoelectric transducer for the generation and full-field
interferometric techniques for the signal detection.
7
However, the aforementioned techniques require a cumber-
some and non-portabl e equipment to control probe positions
or angles of incidence.
In contrast, broadband signals contain more information
than narrowband signals, but the separation of each mode
contribution and the dispersive effects within each mode are
key issues to address and require data processing. Broadband
signals can be acquired with a diversity of techniques, like a
piezoceramic transducer for emitting and a laser scanning
Doppler vibrometer for sensing,
13
laser-ultrasound,
14,15
or
line-focus acoustic microscopy.
16
Spatio-temporal broad-
band signals recorded at equally spaced points are two-
dimensionally (2-D) Fourier transformed to obtain the fre-
quency spectrum of the different propagating modes in the
frequency-wave number plane.
17
Then, for comparison with
theoretical results, this information is generally processed to
extract the ridges of the measured dispersion curves in terms
of frequency-w ave number data pairs. Although scarcely dis-
cussed in the literature, their automatic detection can be
problematic due to mode superposition and limited spatial
resolution.
18
To carefully extract the dispersion curves,
a)
Electronic mail: claire.prada-julia@espci.fr
1138 J. Acoust. Soc. Am. 143 (2), February 2018
V
C
2018 Acoustical Society of America0001-4966/2018/143(2)/1138/10/$30.00
visual operator selection,
19
peak-finding algorithms and
interpolation,
13
image tracing algorithm,
16
matrix pencil
method,
20
and more advanced data processing algorithms, as
reviewed in Refs. 21 and 22, have been developed, often at
the cost of time-consuming procedures.
As a first limitation, most of the former techniques are
designed for the analysis of signals captured in a pointwise
scheme, and the change of the recording location is usually
performed by mechanically displacing the receiver.
Furthermore, measurements are generally carried out in a
propagation plane that coincides with a principal symmetry
plane of the material and the principal symmetry axes and
plate thickness are assumed to be known. Last, to retrieve
elastic properties, least-squares fitting criteria are frequently
applied, i.e., a procedure is followed in order to minimize
the sum of the squared differences between extracted experi-
mental data and theoretical dispersion curves. For this
approach, it is necessary to identify the branches of the dis-
persion curves prior to the inversion process.
5,23
However, it
should be noted that a clear identification of the modes in
experimental data is uncertain, especially for higher-order
modes.
16,19
In this way, Karim et al.
6
presented a numerical
procedure for the inversion of LLW data to determine certain
material properties of a composite laminate using a modified
version of the simplex algorithm. For this approach to be
effective, four broadband transducers were used to record
the phase velocities over a wide frequency range and three
measurements in different propagation directions were
needed to recover the full set of independent elastic coeffi-
cients. In another related study, Yan et al.
16
described a
method for the inversion of the elastic properties of plates by
line-focus acoustic microscopy using a hybrid particle
swarm-based-simulated annealing optimization. However,
this approach required tedious data processing for converting
time-domain waveforms into dispersion curves and was
solely applied to a thin isotropic plate. An alternative user-
independent inversion scheme is thus highly desirable
towards real-time applications.
To face these limitations, the primary objective of the
present work is to develop an effective method to identify
the geometric and anisotropic elastic properties of plate
materials. To this end, an accurate and simple technique to
measure multimode dispersion curves is proposed. This tech-
nique takes advantage of the advent of multi-element probes,
together with multichannel electronics, initially devoted to
medical imaging. Therefore, applications to material charac-
terization are expanding, e.g., for cortical bone assess-
ment
24,25
and structural health monitoring.
26
This approach
offers several advantages: (1) the acquisition procedure is
significantly simpler than that delivered by other setups,
such as the LLW technique and laser-based devices; (2) it
requires minimal data processing and no extraction of the
experimental data; and (3) it is well suited for field use (i.e.,
single-side access, no mechanical displacement). Although
the probe induces leakage, we observe that it does not signif-
icantly modify the phase velocity of the propagating
modes.
27
In addition to the experimental technique, a deci-
sive task is to develop a robust inverse procedure. To avoid
the difficult pairing between experimental data points and
theoretical guided modes, an objective function is bui lt from
the dispersion equation and directly evaluated on experimen-
tal data points. This objective function does not require the
numerical calculation of the theoretical guided modes, and
thus drastically reduces the computational costs for solving
the inverse problem. Besides, we show that the method can
also be applied to simultaneously recover the elastic proper-
ties and the propagation direction angle. Genetic algorithms
are chosen for optimizing the objective function, owing to
their flexibility in solving multiparametric inversion
problem.
28
The reconstruction method is applied to experimental
data measured on materials with various symmetry classes.
Results are presented for isotropic duralumin and fused
quartz plates, a silicon wafer with cubic symmetry and a
transversely isotropic titanium plate. The proposed inversion
procedure is proven feasible to recover the full set of elastic
coefficients, together with the plate thickness or the propaga-
tion plane, from a single dispersion curves measurement.
The paper is structured as follows: the experimental setup
used to measure guided waves is described in Sec. II. This
section also presents the forward model used to predict the
theoretical dispersion curves, along with the inverse proce-
dure employed to retrieve the material properties. The results
obtained for two case studies, i.e., the inference of plate
properties for a known propagation direction and the infer-
ence of elastic properties and propagation angle for a known
plate thickness are exposed in Sec. III. The results are com-
pared to reference values and discussed in the light of com-
putational costs, inverse problem errors, and measurement
uncertainties in Sec. IV.
II. MATERIALS AND METHODS
Measurements of EGWs, along with appropriate model-
ing, have the potential for yielding estimates of waveguide
properties such as thickness, elasticity and the angle of prop-
agation with respect to the principal symmetry axes. Such a
model-based approach requires solving a multiparametric
inverse problem to match the experimental dispersion curves
with the predicted guided modes. The experimental setup
used to measure guided waves in anisotropic plates is first
described. Then, the inverse procedure used to recover mate-
rial properties is presented.
A. Guided waves measureme nts and investigated
samples
EGWs measurements were performed using a commer-
cially available 10-MHz standard probe (Imasonic,
Besanc¸on, France), which consists in a 128-element linear
transducer array. The array pitch was 0.25 mm and the
dimensions of each rectangular element were 0:2 10 mm
2
.
The probe was mounted on a rotational stage (ESP 301
Motion Controller, Newport, CA), allowing measu rements
in any propagation direction / with respect to the principal
symmetry axes of the material. The probe was placed
directly in contact with the sample using tap water for cou-
pling. A Lecoeur Electronics Ltd system was used to trans-
mit a broadband chirp signal at a 8-MHz central frequency
J. Acoust. Soc. Am. 143 (2), February 2018 Bochud et al. 1139
(6 dB power spectrum spanning the frequency range from
4 to 12 MHz) and 4 ls duration to the first element e
E
1
and to
record the received signals on all elements e
R
j
(j ¼ 1; …; 128) during 25 ls. For each propagation direction
/, a set of 128 radio-frequency (RF) signals was digitized
(12 bits, 80 MHz, 8000 samples).
The experimental dispersion curves of the guided modes
were obtained as follows: (1) for each propagation direction /,
the 128 RF signals were Fourier tr ansformed with respect to
time and space,
17
and stored in a response matrix; (2) the mean
noise level (in dB) was evaluated in an ar ea away from the
guided modes (i.e., low phase velocity area), and all components
of the response matrix below this threshold were set to zero; (3)
the remaini ng N non-zero components were stored into a N 3
matrix ½k
n
f
n
w
n
(n ¼ 1; …; N), where k, f,andw stand for
wave number, frequency and Fourier coefficients, respectively.
It is worth mentioning that these data processing steps were
only applied to remove a large part of the background noise and
decrease the amount of data, but the dispersion curves still con-
tain many spurious data (e.g., noise, side lobes). A flowchart of
the acquisition and data processing is depicted in Fig. 1. A video
for this is in cluded as supp lementary material.
29
Measurements were performed on four samples: a 1-
mm thick duralumin plate, a 1.5-mm thick fused quartz
plate, a 0.8-mm thick silicon wafer, and a 1-mm thick tita-
nium plate. Reference thicknesses were determined using a
digital caliper (accuracy of 0.01 mm). The transversal
dimensions of the samples were larger than 50 mm, allowing
the modes to travel along the whole probe array before being
reflected at the plate boundaries. Table I summarizes the
nominal values for the elastic coefficients, the mass density
and the reference thickness of the samples.
B. Estimation of material properties and symmetry
axes
In the following, we introduce the forward waveguide
model used to analyze the elastic wave propagation in aniso-
tropic plates and we shortly recall the main constituent parts
FIG. 1. (Color online) Flowchart of the acquisition and data processing: (a) broadband chirp signal used as excitation; (b) schematic view of the multi-element
probe; (c) radio-frequency signals; (d) dispersion curves after applying a 2-D Fourier transform; (e) dispersion curves after denoising (i.e., the threshold was
set according to the noise level evaluated within the area inside the white box); and (f) final experimental data.
TABLE I. Reference values.
Reference values
Elastic coefficients
(GPa)
Density
(g cm
3
)
Thickness
(mm)
c
11
c
12
c
33
c
13
c
44
q d
Duralumin
a
114.4 – – – 27.2 2.795 0.992
Fused quartz
a
78.4 – – – 31.2 2.200 1.475
Silicon
b
165.6 63.9 – – 79.5 2.329 0.779
Titanium
c
162.4 92.0 180.7 69.0 46.7 4.506 1.012
a
Briggs and Kolosov (Ref. 30).
b
Prada et al. (Ref. 31).
c
Royer et al. (Ref. 32).
1140 J. Acoust. Soc. Am. 143 (2), February 2018 Bochud et al.
